ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS II

Size: px
Start display at page:

Download "ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS II"

Transcription

1 To the memory of my Mother ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS II KRZYSZTOF P LOTKA Abstract. We say that a function h: R R is a Hamel function (h HF) if h, considered as a subset of R 2, is a Hamel basis for R 2. We show that A(HF) ω, i.e., for every finite F R R there exists f R R such that f + F HF. From the previous work of the author it then follows that A(HF) = ω (see [P]). The terminology is standard and follows [C]. The symbols R and Q stand for the sets of all real and all rational numbers, respectively. A basis of R n as a linear space over Q is called Hamel basis. For Y R n, the symbol Lin Q (Y ) stands for the smallest linear subspace of R n over Q that contains Y. The zero element of R n is denoted by 0. All the linear algebra concepts are considered for the field of rational numbers. The cardinality of a set X we denote by X. In particular, c stands for R. Given a cardinal κ, we let cf(κ) denote the cofinality of κ. We say that a cardinal κ is regular if cf(κ) = κ. For any set X, the symbol [X] κ denotes the set {Z X : Z < κ}. For A, B R n, A + B stands for {a + b: a A, b B}. We consider only real-valued functions. No distinction is made between a function and its graph. For any two partial real functions f, g we write f + g, f g for the sum and difference functions defined on dom(f) dom(g). The class of all functions from a set X into a set Y is denoted by Y X. We write f A for the restriction of f Y X to the set A X. For B R n its characteristic function is denoted by χ B. For any function g R X and any family of functions F R X we define g + F = {g + f : f F }. For any planar set P, we denote its x-projection by dom(p ). The cardinal function A(F), for F R X, is defined as the smallest cardinality of a family G R X for which there is no g R X such that g + G F. Recall that f : R n R is a Hamel function (f HF(R n )) if f, considered as a subset of R n+1, is a Hamel basis for R n+1. In [P], it was proved that 3 A(HF(R n )) ω. In the same paper, the author asked whether A(HF(R n )) = ω (Problem 3.5). The following theorem gives a positive answer to this question Mathematics Subject Classification. Primary 26A21, 54C40; Secondary 15A03, 54C30. Key words and phrases. Hamel basis, additive and Hamel functions. The work was supported in part by the intersession research grant from the University of Scranton. 1

2 2 KRZYSZTOF P LOTKA Theorem 1. A(HF(R n )) ω, i.e. for every finite F R Rn, there exists g R Rn such that g + F HF(R n ). Before we prove the theorem we state and prove the following lemmas. Lemma 2. Let b 1,..., b m R be arbitrary numbers. There exists a linear basis C of Lin Q (b 1,..., b m ) such that b i + C is also a basis of Lin Q (b 1,..., b m ), for every i m. Proof. Without loss of generality we may assume that Lin Q (b 1,..., b m ) {0}. Let C = {c 1,..., c k } be any linear basis of Lin Q (b 1,..., b n ). So, for every i m there are p i1,..., p ik Q such that p ij c j = b i. Now, choose q Q \ {0} satisfying the following condition for all i j q j p ij 1. We claim that C = {c 1,..., c k } = 1 q C = { 1 q c 1,..., 1 q c k } is the desired basis. To prove this we need to show that for every i m b i + C is linearly independent. To see this consider a zero linear combination j p ij(b i + c j ) = 0. We have that j p ijc j = b i j p ij. If j p ij = 0 then obviously p i1 = = p ik = 0. So we may assume that j p ij 0. Next we divide both sides of j p ijc j = b i j p ij by j p ij and obtain that p ij j c j pij j = b i. On the other hand j p ij c j = j p ij qc j = b i. So we conclude that p ij j pij = qp ij for all j k and consequently q j p ij = j 1. A contradiction. p ij j pij = Now, since dim(lin Q (b i + C)) = dim(lin Q (C)) and Lin Q (b i + C) Lin Q (C), we conclude that Lin Q (b i + C) = Lin Q (C) = Lin Q (b 1,..., b m ). Let us note here that the above lemma cannot generalized onto infinite case. As a counterexample take {b 1, b 2, b 3,... } = Q and observe that there is no basis C with the required properties. Lemma 3. [PR, Lemma 2] Let H R n be a Hamel basis. Assume that h: R n R is such that h H 0. Then h is a Hamel function iff h (R n \ H) is one-to-one and h[r n \ H] R is a Hamel basis.

3 ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS II 3 Lemma 4. Let X be a set of cardinality c and k 1. TFAE: (a) for all f 1,..., f k : R n R, there exists f R Rn such that f + f i HF(R n ) (i = 1,..., k). (b) for all g 1,..., g k R X, there exists g R X such that g + g i is one-to-one and (g + g i )[X] R is a Hamel basis (i = 1,..., k). Proof. (a) (b) Choose a Hamel basis H R n and a bijection p: R n \ H X. Put f i = (g i p) (0 H). By (a), there exists an f R Rn such that f +f i HF(R n ) (i = 1,..., k). Now, let A Add(R n ) be such that f H = A H and put f = f A. Note that f + f i = (f + f i ) A HF(R n ) Add(R n ) = HF(R n ) (see [P, Fact 3.1]) and also (f + f i ) H 0, (i = 1,..., k). Hence, by Lemma 3 we claim that (f + f i ) (R n \ H) is a bijection onto a Hamel basis. Now define g = f p 1 and note that it is the required function. (b) (a) Let H be like above. Choose A i Add(R n ) such that f i H A i H for every i = 1,..., k. Put X = R n \ H and g i = (f i A i ) X for i = 1,..., k. By (b), there exists a g : X R such that g + g i is a bijection between X and a Hamel basis. Define f = g (0 H) and observe that f + f i = [f + (f i A i )] + A i = [(g + g i ) (0 H)] + A i. Since (g + g i ) (0 H) HF(R n ) by Lemma 3, using [P, Fact 3.1] we conclude that [(g + g i ) (0 H)] + A i HF(R n ) for each i = 1,..., k. Hence f is the required function. Lemma 5. Let X be a set of cardinality c, ω κ < c, and f 1,... f k be functions such that f i [X] = c. R X Then there exist pairwise disjoint subsets A 1,... A n X of cardinality κ + each and satisfying the following property: for every i = 1,..., k and j = 1,..., n the restriction f i A j is one-to-one or constant and f i [ A j ] = κ + (i.e. f i is one-to-one on at least one of the sets). Proof. We prove the lemma by induction on k. If k = 1, then the conclusion is obvious (note that κ + c). Now assume that the lemma holds for k ω and let f 1,... f k+1 R X be functions such that f i [X] = c. Based on the inductive assumption, let A 1,... A n X be sets with the required properties for the functions f 1,... f k. If f k+1 [ A i ] = κ +, then by reducing the original sets A 1,... A n we will obtain sets which work for all the functions f 1,... f k+1. In the case when f k+1 [ A i ] κ, we can find a subset A n+1 X disjoint with n 1 A i such that A n+1 = κ + and f k+1 A n+1 is injective. Now, by appropriately reducing the sets A 1,... A n+1 we will obtain the desired sets.

4 4 KRZYSZTOF P LOTKA Lemma 6. Let X be a set of cardinality c, f 1,... f k R X be functions such that f i [X] = c, B 0, B 1 R be such that B 0 B 1 < c, and y R \ Lin Q (B 0 ). Then there exist y 1,..., y n R and x 1,..., x n X such that (a) n 1 y j = y, (b) {y 1,..., y n }, {y j + f i (x j ): j = 1,..., n} are both linearly independent over Q and Lin Q ({y 1,..., y n }) Lin Q (B 0 ) = Lin Q ({y j + f i (x j ): j = 1,..., n}) Lin Q (B 1 ) = {0} for all i = 1,..., k. Proof. Put κ = B 0 B 1 ω and let A 1,... A n X be the sets from Lemma 5 for functions f 1,... f k. First we will define the values y 1,..., y n. Let {b 1,... b s } be the set of all values such that f i A j b l for some i, j, l. Choose y 2,..., y n to be linearly independent over Q such that Lin Q ({y 2,..., y n }) Lin Q (B 0 B 1 {b 1,..., b s, y}) = {0}. This can be easily done by extending the basis of Lin Q (B 0 B 1 {b 1,..., b s, y}) to a Hamel basis and selecting (n 1) elements from the extension. Next define y 1 = y (y y n ). Obviously n 1 y j = y. We claim that {y 1,..., y n } is linearly independent over Q and Lin Q ({y 1,..., y n }) Lin Q (B 0 ) = {0}. Assume that α 1 y α n y n = 0 for some rationals α 1,..., α n. From the definition of y 1 we get (α 2 α 1 )y 2 + +(α n α 1 )y n = α 1 y. Based on the way y 2,..., y n were selected we conclude that α 1 = 0 and consequently α 2 = = α n = 0. Next assume that q 1 y q n y n = b for some rationals q 1,..., q n and b Lin Q (B 0 ). above, we obtain that (q 2 q 1 )y (q n q 1 )y n Then, proceeding similarly like Lin Q (B 0 {y}), which implies that q 1 = = q n. Consequently, if q 1 0, then we could conclude that y Lin Q (B 0 ). That would contradict one of the assumptions of the lemma. Hence q 1 = = q n = 0 and the sequence y 1,..., y n satisfies the required conditions. Before we define the sequence x 1,..., x n, we observe some additional properties of y 1,..., y n. Fix 1 i k. Let A i1,..., A il (i 1 < < i l ) be all the sets on which f i is constant and let b i1,..., b il be the values of f i on these sets, respectively. Note that properties of the sets A 1,..., A n imply that {i 1,..., i l } {1,..., n}. We will show that (1) (y i1 + b i1 ),..., (y il + b il ) are linearly independent, (2) Lin Q ({(y i1 + b i1 ),..., (y il + b il )}) Lin Q (B 1 ) = {0}. To see (1) assume that α 1 (y i1 + b i1 ) + + α l (y il + b il ) = 0 for some rationals α 1,..., α l. This implies α 1 y i1 + + α l y il = (α 1 b i1 + + α l b il ) Lin Q (B 0 B 1 {b 1,..., b s, y}). If i 1 1, then it easily follows that α 1 = = α l = 0. If i 1 = 1, then we can write α 1 y i1 + + α l y il = α 1 y 1 + α 2 y i2 + + α l y il = α 1 [y (y y n )] + α 2 y i2 + + α l y il Lin Q (B 0 B 1 {b 1,..., b s, y}).

5 ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS II 5 Consequently, α 1 (y 2 + +y n )+α 2 y i2 + +α l y il Lin Q (B 0 B 1 {b 1,..., b s, y}). Since {i 1,..., i l } {1,..., n}, after simplifying the expression α 1 (y y n ) + α 2 y i2 + + α l y il, there will be at least one term y j with the coefficient being exactly α 1. Hence, we conclude that α 1 = 0 and as a consequence of that α 2 = = α l = 0. This finishes the proof of (1). Similar argument shows (2). Next we will define the elements x 1,..., x n X (by induction). Choose x 1 A 1 \ f 1 i [Lin Q (B 1 {b 1, b 2,..., b s, y 1,..., y n })]. i k,f i is 1-1 on A 1 This choice is possible since A 1 = κ + > κ Lin Q (B 1 {b 1, b 2,..., b s, y 1,..., y n }) and together with the condition (2) assures that Lin Q ({y 1 + f i (x 1 ): f i A 1 is 1-1}) Lin Q (B 1 {b 1, b 2,..., b s, y 1,..., y n }) {0} and Lin Q ({y 1 + f i (x 1 )}) Lin Q (B 1 ) = {0} for all i k. Now assume that x 1 A 1,..., x m 1 A m 1 (m < n) have been defined and they satisfy the following property: ( ) {y j + f i (x j ): j = 1,..., m 1} is linearly independent, Lin Q ({y j + f i (x j ): j m 1 and f i A j is 1-1}) Lin Q (B 1 {b 1, b 2,..., b s, y 1,..., y n }) {0}, and Lin Q ({y j + f i (x j ): j = 1,..., m 1}) Lin Q (B 1 ) = {0} for all i = 1,..., k. Choose x m A m such that x m i k,f i is 1-1 on A m The choice of x m implies that f 1 i [Lin Q (B 1 {b 1, b 2,..., b s, y 1,..., y n, f i (x 1 ),..., f i (x m 1 )})]. y m + f i (x m ) Lin Q (B 1 {b 1, b 2,..., b s, y 1,..., y n, f i (x 1 ),..., f i (x m 1 )}) for all i k such that f i is 1-1 on A m. This combined with the inductive assumption ( ) and the conditions (1) and (2) leads to the conclusion that {y j +f i (x j ): j = 1,..., m} is linearly independent, Lin Q ({y j + f i (x j ): j m and f i A j is 1-1}) Lin Q (B 1 {b 1, b 2,..., b s, y 1,..., y n }) {0}, and Lin Q ({y j +f i (x j ): j = 1,..., m}) Lin Q (B 1 ) = {0} for all i = 1,..., k. Based on the induction we claim that the sequence x 1,..., x n X has been constructed and it satisfies the following condition: {y j + f i (x j ): j = 1,..., n} is linearly independent and Lin Q ({y j + f i (x j ): j = 1,..., n}) Lin Q (B 1 ) = {0} for all i = 1,..., k. Summarizing, the sequences x 1,..., x n X and y 1,..., y n R satisfying the conditions (a) and (b) have been constructed. Remark 7. Let A A and f 1, f 2 : A R. If (f 1 f 2 )[A] Lin Q (f 1 [A ]) Lin Q (f 2 [A ]), then Lin Q (f 1 [A]) = Lin Q (f 2 [A]).

6 6 KRZYSZTOF P LOTKA The remark easily follows from the equality l α i f 1 (x i ) = 1 l α i f 2 (x i ) + 1 l α i [f 1 (x i ) f 2 (x i )]. Proof of Theorem 1. Let X be a set of cardinality c. By Lemma 4, it suffices to show that for arbitrary f 1,..., f k : X R there exists a function g : X R such that g + f i is 1-1 and (g + f i )[X] is a Hamel basis (i = 1,..., k). The proof in the general case will be by transfinite induction with the use of the previously stated auxiliary results. However, in the special case when f i [X] < c for all i, it can be presented without the use of induction. The method is interesting and also used in part of the proof of general case, so we present it here. Assume that f i [X] < c for all i, let V = Lin Q ( f i [X]), and λ < c be the cardinality of a linear basis of V. 1 Choose Z X such that Z = λ and f i Z is a constant function for every i and let {b 1,..., b m } = f i [Z]. Next we define a Hamel basis H. Let C be a basis of Lin Q (b 1,..., b m ) from Lemma 2, H 1 be an extension of C to a basis of V, and finally H be an extension of H 1 to a Hamel basis. Define g : X H as a bijection with the property that g[z] = H 1. We claim that g + f i is 1-1 and (g + f i )[X] is a Hamel basis (i = 1,..., k). To see this recall that b j + C is linearly independent, Lin Q (b j + C) = Lin Q (C) = Lin Q ({b 1,..., b m }) (see Lemma 2), and C H 1. This implies that Lin Q (b j +H 1 ) = Lin Q (H 1 ), b j +(H 1 \C) is linearly independent, and as a consequence, b j + H 1 is linearly independent. Therefore, since f i [Z] = {b j } for some j, we have that (g + f i )[Z] = b j + H 1. Thus (g + f i )[Z] is linearly independent and Lin Q ((g + f i )[Z]) = Lin Q (H 1 ). Finally, since f i [X] Lin Q (H 1 ) = Lin Q ((g + f i )[Z]), we can similarly conclude that (g + f i )[X] is linearly independent and Lin Q ((g + f i )[X]) = Lin Q (g[x]) = Lin Q (g[x]) = R. This finishes the proof of the special case. Now we prove the result for arbitrary functions f 1,..., f k : X R. We start by dividing {f 1,..., f k } into abstract classes according to the relation: f i f j iff (f i f j )[X] < c (it is easy to verify that this is an equivalence relation). Put K = i f j f i (f i f j )[X], κ = ω K, and note that κ < c. There exists a set Z X such that Z = κ + and for all i, j the function (f i f j ) Z is one-to-one or constant (the existence of such a set can be shown by using an argument similar to the one from the proof of Lemma 5; obviously, if f i f j, then (f i f j ) Z is constant). Our goal is to define g : Z R for some Z Z such that for every i k g + f i is injective, (g + f i )[Z ] is linearly independent, and K Lin Q ((g + f i )[Z ]). Define V = Lin Q (K) and introduce another equivalence relation among the functions f 1,..., f k : f i = fj iff (f i f j ) Z is constant. Note that =. Let

7 ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS II 7 f i1,..., f il be representatives of the abstract classes of the relation =. Consider l s=1 f j =fis (f j f is )[Z] = {b 1,..., b m }. By Lemma 2, there exists a linear basis C of Lin Q ({b 1,..., b m }) such that b r + C (r m) is also a linear basis for Lin Q ({b 1,..., b m }). Let H 1 be a linear basis of V extending C. Choose a set Z 1 Z such that Z 1 = H 1 and (f j f i1 )[Z 1 ] is linearly independent and Lin Q ((f j f i1 )[Z 1 ]) V = {0} for all f j = fi1. This can be done since Z = κ + > V H 1 and (f j f i1 ) Z is injective for every f j = fi1. Let g 1 : Z 1 H 1 be a bijection and define g : Z 1 R by g = g 1 f i1. Then g+f j is oneto-one for all j, (g + f j )[Z 1 ] is linearly independent for all j, Lin Q ((g + f j )[Z 1 ]) = V for f j = fi1 (see the argument in the special case in the beginning of the proof), and Lin Q ((g + f j )[Z 1 ]) V = {0} for f j = fi1 (the latter follows from the fact that if Y 1 and Y 2 are both linearly independent and Lin Q (Y 1 ) Lin Q (Y 2 ) = {0}, then Y 1 +Y 2 is also linearly independent and Lin Q (Y 1 ) Lin Q (Y 1 +Y 2 ) = Lin Q (Y 1 ) Lin Q (Y 1 +Y 2 ) = {0}). Next choose a set Z 2 Z \ Z 1 such that Z 2 = H 1, (f j f i2 )[Z 2 ] is linearly independent, and Lin Q ((f j f i2 )[Z 2 ]) Lin Q ( k 1 (g + f i)[z 1 ]) = {0} for all f j = fi2 (note that V k 1 (g+f i)[z 1 ] since Lin Q ((g+f i1 )[Z 1 ]) = V ). This choice is possible for similar reasons as in the case of Z 1. Let g 2 : Z 2 H 1 be a bijection and extend g onto Z 1 Z 2 by defining it on Z 2 as g = g 2 f i2. Then g + f j is one-to-one for all j, (g + f j )[Z 1 Z 2 ] is linearly independent for all j, V Lin Q ((g + f j )[Z 1 Z 2 ]) for f j = fi1 or f j = fi2, and Lin Q ((g + f j )[Z 1 Z 2 ]) V = {0} for f j = fi1 and f j = fi2. By continuing this process (or more formally, by using the mathematical induction), we construct a sequence of pairwise disjoint sets Z 1, Z 2,..., Z l Z and a partial real function g : Z R (Z = Z 1 Z l ) such that for each j = 1,..., k, g +f j is one-to-one, (g +f j )[Z ] is linearly independent, and V Lin Q ((g +f j )[Z ]). Observe also that Z κ. Therefore X \ Z = c. In the following part of the proof, we will use the transfinite induction to extend the partial function g onto the whole set X making sure it possesses the desired properties. We will make use of Lemma 6 and Remark 7. First notice that if Z A X and g : A R is any extension of g : Z R then for f j f i we have that ((g + f j ) (g + f i ))[A] = (f j f i )[A] (f j f i )[X] V Lin Q ((g + f i )[Z ]) Lin Q ((g + f j )[Z ]). Hence the remark implies that Lin Q ((g + f i )[A]) = Lin Q ((g + f j )[A]). Thus, when extending the function g it will suffice to consider only the representatives of the abstract classes of the relation. Let f j1,..., f jt be those functions. Let H = {h ξ : ξ < c} be a Hamel basis and {x ξ : ξ < c} be an

8 8 KRZYSZTOF P LOTKA enumeration of X \ Z. We will define a sequence of pairwise disjoint finite sets {X ξ : ξ < c} such that ξ<c X ξ = X \ Z, x ξ β ξ X β and an extension of g onto X such that for each ξ < c the following condition holds (P ξ ) g + f jr is one-to-one, (g + f jr )[Z β ξ X β] is linearly independent, and h ξ Lin Q ((g + f jr )[Z β ξ X β]) for all r = 1,..., t. Notice that this will finish the proof of our main theorem. To perform the inductive construction, fix α < c and assume that the sets X ξ have been defined for all ξ < α and the function g extended onto Z ξ<α X ξ in such a way that (P ξ ) is satisfied for each ξ < α. If x α Z ξ<α X ξ, then define g(x α ) t r=1 Lin Q((g + f jr )[Z ξ<α X ξ] {f jr (x α )}). This assures that g+f jr is one-to-one and (g+f jr )[Z ξ<α X ξ {x α }] is linearly independent (r = 1,..., t). Next, if h α (g + f j1 )[Z ξ<α X ξ {x α }], then put X α1 =. Otherwise, we apply Lemma 6 to functions f jr f j1 : X \ (Z ξ<α X ξ {x α }) R (r = 2,..., t), B 0 = Lin Q ((g + f j1 )[Z ξ<α X ξ {x α }]), B 1 = Lin Q ( t r=2 (g + f j r )[Z ξ<α X ξ {x α }]), and y = h α. Hence there exist y 1j1,..., y n1j 1 R and x 1j1,..., x n1j 1 X\(Z ξ<α X ξ {x α }) such that the conditions (a) and (b) from the lemma are satisfied. We define X α1 = {x 1j1,..., x n1j 1 } and g(x ij1 ) = y ijr f j1 (x ij1 ) (i = 1..., n 1 ). By repeating the above steps for the other functions f j2,..., f jt (the sets B 0 and B 1 need to be appropriately extended in each step) we obtain pairwise disjoint sets X α1,..., X αt X \(Z ξ<α X ξ {x α } and an extension of g onto Z ξ α X ξ (where X α = X α1 X αt {x α }). Observe that the conditions (a) and (b) from Lemma 6 imply that (P α ) holds. This completes the inductive construction and also the proof of Theorem 1. References [CA] A.L Cauchy, Cours d analyse de l Ecole Polytechnique, 1. Analyse algébrique, V., Paris, 1821 [Oeuvres (2) 3, Paris, 1897]. [C] K. Ciesielski, Set Theory for the Working Mathematician, London Math. Soc. Student Texts 39, Cambridge Univ. Press [CM] K. Ciesielski, A.W. Miller Cardinal invariants concerning functions whose sum is almost continuous, Real Anal. Exchange 20 ( ), [CN] K. Ciesielski, T. Natkaniec, Algebraic properties of the class of Sierpiński-Zygmund functions, Topology Appl. 79 (1997), [CR] K. Ciesielski, I. Rec law, Cardinal invariants concerning extendable and peripherally continuous functions, Real Anal. Exchange 21 ( ), [H] G. Hamel, Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung f(x + y) = f(x) + f(y), Math. Ann. 60 (1905), [HW] W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, [MK] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Publishers, PWN, Warszawa, [P] K. P lotka, On functions whose graph is a Hamel basis, Proc. Amer. Math. Soc. 131 (2003),

9 ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS II 9 [PR] K. P lotka, I. Rec law, Finitely Continuous Hamel Functions, Real Anal. Exchange, 30(2) ( ), 1 4. Department of Mathematics, University of Scranton, Scranton, PA 18510, USA address: Krzysztof.Plotka@scranton.edu

ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS

ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS ON FUNCTIONS WHOSE GRAPH IS A HAMEL BASIS KRZYSZTOF P LOTKA Abstract. We say that a function h: R R is a Hamel function (h HF) if h, considered as a subset of R 2, is a Hamel basis for R 2. We prove that

More information

SOME ADDITIVE DARBOUX LIKE FUNCTIONS

SOME ADDITIVE DARBOUX LIKE FUNCTIONS Journal of Applied Analysis Vol. 4, No. 1 (1998), pp. 43 51 SOME ADDITIVE DARBOUX LIKE FUNCTIONS K. CIESIELSKI Received June 11, 1997 and, in revised form, September 16, 1997 Abstract. In this note we

More information

The ideal of Sierpiński-Zygmund sets on the plane

The ideal of Sierpiński-Zygmund sets on the plane The ideal of Sierpiński-Zygmund sets on the plane Krzysztof P lotka Department of Mathematics, West Virginia University Morgantown, WV 26506-6310, USA kplotka@math.wvu.edu and Institute of Mathematics,

More information

On sums of Darboux and nowhere constant continuous functions

On sums of Darboux and nowhere constant continuous functions On sums of Darboux and nowhere constant continuous functions Krzysztof Ciesielski Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA e-mail: K Cies@math.wvu.edu; web page:

More information

A Primer on the Functional Equation

A Primer on the Functional Equation A Primer on the Functional Equation f (x + y) = f (x) + f (y) Kaushal Verma Let f : R R be a function satisfying f (x + y) = f (x) + f (y), (0.1) for all x, y R. Historical records at the École Polytechnique

More information

CARDINAL INVARIANTS CONCERNING EXTENDABLE AND PERIPHERALLY CONTINUOUS FUNCTIONS

CARDINAL INVARIANTS CONCERNING EXTENDABLE AND PERIPHERALLY CONTINUOUS FUNCTIONS RESEARCH Real Analysis Exchange Vol. 21(2), 1995 96, pp. 459 472 Krzysztof Ciesielski, Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, e-mail: kcies@@wvnvms.wvnet.edu Ireneusz

More information

SMALL COMBINATORIAL CARDINAL CHARACTERISTICS AND THEOREMS OF EGOROV AND BLUMBERG

SMALL COMBINATORIAL CARDINAL CHARACTERISTICS AND THEOREMS OF EGOROV AND BLUMBERG Real Analysis Exchange Vol. 26(2), 2000/2001, pp. 905 911 Krzysztof Ciesielski, Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA, email: K Cies@math.wvu.edu, internet:

More information

ALGEBRAIC SUMS OF SETS IN MARCZEWSKI BURSTIN ALGEBRAS

ALGEBRAIC SUMS OF SETS IN MARCZEWSKI BURSTIN ALGEBRAS François G. Dorais, Department of Mathematics, Dartmouth College, 6188 Bradley Hall, Hanover, NH 03755, USA (e-mail: francois.g.dorais@dartmouth.edu) Rafa l Filipów, Institute of Mathematics, University

More information

Jónsson posets and unary Jónsson algebras

Jónsson posets and unary Jónsson algebras Jónsson posets and unary Jónsson algebras Keith A. Kearnes and Greg Oman Abstract. We show that if P is an infinite poset whose proper order ideals have cardinality strictly less than P, and κ is a cardinal

More information

Uncountable γ-sets under axiom CPA game

Uncountable γ-sets under axiom CPA game F U N D A M E N T A MATHEMATICAE 176 (2003) Uncountable γ-sets under axiom CPA game by Krzysztof Ciesielski (Morgantown, WV), Andrés Millán (Morgantown, WV) and Janusz Pawlikowski (Wrocław) Abstract. We

More information

Smoothness and uniqueness in ridge function representation

Smoothness and uniqueness in ridge function representation Available online at www.sciencedirect.com ScienceDirect Indagationes Mathematicae 24 (2013) 725 738 www.elsevier.com/locate/indag Smoothness and uniqueness in ridge function representation A. Pinkus Department

More information

Composition of Axial Functions of Product of Finite Sets

Composition of Axial Functions of Product of Finite Sets Composition of Axial Functions of Product of Finite Sets Krzysztof P lotka Department of Mathematics University of Scranton Scranton, PA 18510, USA plotkak2@scranton.edu September 14, 2004 Abstract To

More information

20 Ordinals. Definition A set α is an ordinal iff: (i) α is transitive; and. (ii) α is linearly ordered by. Example 20.2.

20 Ordinals. Definition A set α is an ordinal iff: (i) α is transitive; and. (ii) α is linearly ordered by. Example 20.2. 20 Definition 20.1. A set α is an ordinal iff: (i) α is transitive; and (ii) α is linearly ordered by. Example 20.2. (a) Each natural number n is an ordinal. (b) ω is an ordinal. (a) ω {ω} is an ordinal.

More information

THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS. K. R. Goodearl and E. S. Letzter

THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS. K. R. Goodearl and E. S. Letzter THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS K. R. Goodearl and E. S. Letzter Abstract. In previous work, the second author introduced a topology, for spaces of irreducible representations,

More information

Exercises for Unit VI (Infinite constructions in set theory)

Exercises for Unit VI (Infinite constructions in set theory) Exercises for Unit VI (Infinite constructions in set theory) VI.1 : Indexed families and set theoretic operations (Halmos, 4, 8 9; Lipschutz, 5.3 5.4) Lipschutz : 5.3 5.6, 5.29 5.32, 9.14 1. Generalize

More information

Axioms of separation

Axioms of separation Axioms of separation These notes discuss the same topic as Sections 31, 32, 33, 34, 35, and also 7, 10 of Munkres book. Some notions (hereditarily normal, perfectly normal, collectionwise normal, monotonically

More information

Math 455 Some notes on Cardinality and Transfinite Induction

Math 455 Some notes on Cardinality and Transfinite Induction Math 455 Some notes on Cardinality and Transfinite Induction (David Ross, UH-Manoa Dept. of Mathematics) 1 Cardinality Recall the following notions: function, relation, one-to-one, onto, on-to-one correspondence,

More information

Measurable Envelopes, Hausdorff Measures and Sierpiński Sets

Measurable Envelopes, Hausdorff Measures and Sierpiński Sets arxiv:1109.5305v1 [math.ca] 24 Sep 2011 Measurable Envelopes, Hausdorff Measures and Sierpiński Sets Márton Elekes Department of Analysis, Eötvös Loránd University Budapest, Pázmány Péter sétány 1/c, 1117,

More information

ON DENSITY TOPOLOGIES WITH RESPECT

ON DENSITY TOPOLOGIES WITH RESPECT Journal of Applied Analysis Vol. 8, No. 2 (2002), pp. 201 219 ON DENSITY TOPOLOGIES WITH RESPECT TO INVARIANT σ-ideals J. HEJDUK Received June 13, 2001 and, in revised form, December 17, 2001 Abstract.

More information

A CRITERION FOR THE EXISTENCE OF TRANSVERSALS OF SET SYSTEMS

A CRITERION FOR THE EXISTENCE OF TRANSVERSALS OF SET SYSTEMS A CRITERION FOR THE EXISTENCE OF TRANSVERSALS OF SET SYSTEMS JERZY WOJCIECHOWSKI Abstract. Nash-Williams [6] formulated a condition that is necessary and sufficient for a countable family A = (A i ) i

More information

van Rooij, Schikhof: A Second Course on Real Functions

van Rooij, Schikhof: A Second Course on Real Functions vanrooijschikhof.tex April 25, 2018 van Rooij, Schikhof: A Second Course on Real Functions Notes from [vrs]. Introduction A monotone function is Riemann integrable. A continuous function is Riemann integrable.

More information

6 CARDINALITY OF SETS

6 CARDINALITY OF SETS 6 CARDINALITY OF SETS MATH10111 - Foundations of Pure Mathematics We all have an idea of what it means to count a finite collection of objects, but we must be careful to define rigorously what it means

More information

En busca de la linealidad en Matemáticas

En busca de la linealidad en Matemáticas En busca de la linealidad en Matemáticas M. Cueto, E. Gómez, J. Llorente, E. Martínez, D. Rodríguez, E. Sáez VIII Escuela taller de Análisis Funcional Bilbao, 9 Marzo 2018 1 Motivation 2 Everywhere differentiable

More information

arxiv:math/ v1 [math.ho] 2 Feb 2005

arxiv:math/ v1 [math.ho] 2 Feb 2005 arxiv:math/0502049v [math.ho] 2 Feb 2005 Looking through newly to the amazing irrationals Pratip Chakraborty University of Kaiserslautern Kaiserslautern, DE 67663 chakrabo@mathematik.uni-kl.de 4/2/2004.

More information

Reverse mathematics and marriage problems with unique solutions

Reverse mathematics and marriage problems with unique solutions Reverse mathematics and marriage problems with unique solutions Jeffry L. Hirst and Noah A. Hughes January 28, 2014 Abstract We analyze the logical strength of theorems on marriage problems with unique

More information

SPACES ENDOWED WITH A GRAPH AND APPLICATIONS. Mina Dinarvand. 1. Introduction

SPACES ENDOWED WITH A GRAPH AND APPLICATIONS. Mina Dinarvand. 1. Introduction MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69, 1 (2017), 23 38 March 2017 research paper originalni nauqni rad FIXED POINT RESULTS FOR (ϕ, ψ)-contractions IN METRIC SPACES ENDOWED WITH A GRAPH AND APPLICATIONS

More information

MATH FINAL EXAM REVIEW HINTS

MATH FINAL EXAM REVIEW HINTS MATH 109 - FINAL EXAM REVIEW HINTS Answer: Answer: 1. Cardinality (1) Let a < b be two real numbers and define f : (0, 1) (a, b) by f(t) = (1 t)a + tb. (a) Prove that f is a bijection. (b) Prove that any

More information

A BOREL SOLUTION TO THE HORN-TARSKI PROBLEM. MSC 2000: 03E05, 03E20, 06A10 Keywords: Chain Conditions, Boolean Algebras.

A BOREL SOLUTION TO THE HORN-TARSKI PROBLEM. MSC 2000: 03E05, 03E20, 06A10 Keywords: Chain Conditions, Boolean Algebras. A BOREL SOLUTION TO THE HORN-TARSKI PROBLEM STEVO TODORCEVIC Abstract. We describe a Borel poset satisfying the σ-finite chain condition but failing to satisfy the σ-bounded chain condition. MSC 2000:

More information

Topological automorphisms of modified Sierpiński gaskets realize arbitrary finite permutation groups

Topological automorphisms of modified Sierpiński gaskets realize arbitrary finite permutation groups Topology and its Applications 101 (2000) 137 142 Topological automorphisms of modified Sierpiński gaskets realize arbitrary finite permutation groups Reinhard Winkler 1 Institut für Algebra und Diskrete

More information

A CHARACTERIZATION OF POWER HOMOGENEITY G. J. RIDDERBOS

A CHARACTERIZATION OF POWER HOMOGENEITY G. J. RIDDERBOS A CHARACTERIZATION OF POWER HOMOGENEITY G. J. RIDDERBOS Abstract. We prove that every -power homogeneous space is power homogeneous. This answers a question of the author and it provides a characterization

More information

Darboux functions. Beniamin Bogoşel

Darboux functions. Beniamin Bogoşel Darboux functions Beniamin Bogoşel Abstract The article presents the construction of some real functions with surprising properties, including a discontinuous solution for the Cauchy functional equation

More information

Ultrafilters with property (s)

Ultrafilters with property (s) Ultrafilters with property (s) Arnold W. Miller 1 Abstract A set X 2 ω has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P 2 ω there exists a perfect set Q P such that Q X or Q X =. Suppose

More information

ON SOME PROPERTIES OF ESSENTIAL DARBOUX RINGS OF REAL FUNCTIONS DEFINED ON TOPOLOGICAL SPACES

ON SOME PROPERTIES OF ESSENTIAL DARBOUX RINGS OF REAL FUNCTIONS DEFINED ON TOPOLOGICAL SPACES Real Analysis Exchange Vol. 30(2), 2004/2005, pp. 495 506 Ryszard J. Pawlak and Ewa Korczak, Faculty of Mathematics, Lódź University, Banacha 22, 90-238 Lódź, Poland. email: rpawlak@imul.uni.lodz.pl and

More information

SMALL SUBSETS OF THE REALS AND TREE FORCING NOTIONS

SMALL SUBSETS OF THE REALS AND TREE FORCING NOTIONS SMALL SUBSETS OF THE REALS AND TREE FORCING NOTIONS MARCIN KYSIAK AND TOMASZ WEISS Abstract. We discuss the question which properties of smallness in the sense of measure and category (e.g. being a universally

More information

ON THE UNIQUENESS PROPERTY FOR PRODUCTS OF SYMMETRIC INVARIANT PROBABILITY MEASURES

ON THE UNIQUENESS PROPERTY FOR PRODUCTS OF SYMMETRIC INVARIANT PROBABILITY MEASURES Georgian Mathematical Journal Volume 9 (2002), Number 1, 75 82 ON THE UNIQUENESS PROPERTY FOR PRODUCTS OF SYMMETRIC INVARIANT PROBABILITY MEASURES A. KHARAZISHVILI Abstract. Two symmetric invariant probability

More information

CERTAIN WEAKLY GENERATED NONCOMPACT, PSEUDO-COMPACT TOPOLOGIES ON TYCHONOFF CUBES. Leonard R. Rubin University of Oklahoma, USA

CERTAIN WEAKLY GENERATED NONCOMPACT, PSEUDO-COMPACT TOPOLOGIES ON TYCHONOFF CUBES. Leonard R. Rubin University of Oklahoma, USA GLASNIK MATEMATIČKI Vol. 51(71)(2016), 447 452 CERTAIN WEAKLY GENERATED NONCOMPACT, PSEUDO-COMPACT TOPOLOGIES ON TYCHONOFF CUBES Leonard R. Rubin University of Oklahoma, USA Abstract. Given an uncountable

More information

Mathematics 220 Workshop Cardinality. Some harder problems on cardinality.

Mathematics 220 Workshop Cardinality. Some harder problems on cardinality. Some harder problems on cardinality. These are two series of problems with specific goals: the first goal is to prove that the cardinality of the set of irrational numbers is continuum, and the second

More information

Topology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.

Topology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved. Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124

More information

REMARKS ON THE SCHAUDER TYCHONOFF FIXED POINT THEOREM

REMARKS ON THE SCHAUDER TYCHONOFF FIXED POINT THEOREM Vietnam Journal of Mathematics 28 (2000) 127 132 REMARKS ON THE SCHAUDER TYCHONOFF FIXED POINT THEOREM Sehie Park 1 and Do Hong Tan 2 1. Seoul National University, Seoul, Korea 2.Institute of Mathematics,

More information

CONTINUOUS EXTENSIONS OF FUNCTIONS DEFINED ON SUBSETS OF PRODUCTS WITH

CONTINUOUS EXTENSIONS OF FUNCTIONS DEFINED ON SUBSETS OF PRODUCTS WITH CONTINUOUS EXTENSIONS OF FUNCTIONS DEFINED ON SUBSETS OF PRODUCTS WITH THE κ-box TOPOLOGY W. W. COMFORT, IVAN S. GOTCHEV, AND DOUGLAS RIVERS Abstract. Consider these results: (a) [N. Noble, 1972] every

More information

Differentiability on perfect subsets P of R; Smooth Peano functions from P onto P 2

Differentiability on perfect subsets P of R; Smooth Peano functions from P onto P 2 Differentiability on perfect subsets P of R; Smooth Peano functions from P onto P 2 Krzysztof Chris Ciesielski Department of Mathematics, West Virginia University and MIPG, Departmentof Radiology, University

More information

Stat 451: Solutions to Assignment #1

Stat 451: Solutions to Assignment #1 Stat 451: Solutions to Assignment #1 2.1) By definition, 2 Ω is the set of all subsets of Ω. Therefore, to show that 2 Ω is a σ-algebra we must show that the conditions of the definition σ-algebra are

More information

A G δ IDEAL OF COMPACT SETS STRICTLY ABOVE THE NOWHERE DENSE IDEAL IN THE TUKEY ORDER

A G δ IDEAL OF COMPACT SETS STRICTLY ABOVE THE NOWHERE DENSE IDEAL IN THE TUKEY ORDER A G δ IDEAL OF COMPACT SETS STRICTLY ABOVE THE NOWHERE DENSE IDEAL IN THE TUKEY ORDER JUSTIN TATCH MOORE AND S LAWOMIR SOLECKI Abstract. We prove that there is a G δ σ-ideal of compact sets which is strictly

More information

A NOTE ON THE EIGHTFOLD WAY

A NOTE ON THE EIGHTFOLD WAY A NOTE ON THE EIGHTFOLD WAY THOMAS GILTON AND JOHN KRUEGER Abstract. Assuming the existence of a Mahlo cardinal, we construct a model in which there exists an ω 2 -Aronszajn tree, the ω 1 -approachability

More information

ON BASIC EMBEDDINGS OF COMPACTA INTO THE PLANE

ON BASIC EMBEDDINGS OF COMPACTA INTO THE PLANE ON BASIC EMBEDDINGS OF COMPACTA INTO THE PLANE Abstract. A compactum K R 2 is said to be basically embedded in R 2 if for each continuous function f : K R there exist continuous functions g, h : R R such

More information

Topology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski

Topology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Topology, Math 581, Fall 2017 last updated: November 24, 2017 1 Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Class of August 17: Course and syllabus overview. Topology

More information

Homogeneity and rigidity in Erdős spaces

Homogeneity and rigidity in Erdős spaces Comment.Math.Univ.Carolin. 59,4(2018) 495 501 495 Homogeneity and rigidity in Erdős spaces KlaasP.Hart, JanvanMill To the memory of Bohuslav Balcar Abstract. The classical Erdős spaces are obtained as

More information

5 Set Operations, Functions, and Counting

5 Set Operations, Functions, and Counting 5 Set Operations, Functions, and Counting Let N denote the positive integers, N 0 := N {0} be the non-negative integers and Z = N 0 ( N) the positive and negative integers including 0, Q the rational numbers,

More information

Closure properties of function spaces

Closure properties of function spaces @ Applied General Topology c Universidad Politécnica de Valencia Volume 4, No. 2, 2003 pp. 255 261 Closure properties of function spaces Ljubiša D.R. Kočinac Dedicated to Professor S. Naimpally on the

More information

ON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES

ON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES Commun. Korean Math. Soc. 22 (2007), No. 2, pp. 297 303 ON SPACES IN WHICH COMPACT-LIKE SETS ARE CLOSED, AND RELATED SPACES Woo Chorl Hong Reprinted from the Communications of the Korean Mathematical Society

More information

Topology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.

Topology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved. Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124

More information

Fixed point of ϕ-contraction in metric spaces endowed with a graph

Fixed point of ϕ-contraction in metric spaces endowed with a graph Annals of the University of Craiova, Mathematics and Computer Science Series Volume 374, 2010, Pages 85 92 ISSN: 1223-6934 Fixed point of ϕ-contraction in metric spaces endowed with a graph Florin Bojor

More information

Disjointness conditions in free products of. distributive lattices: An application of Ramsay's theorem. Harry Lakser< 1)

Disjointness conditions in free products of. distributive lattices: An application of Ramsay's theorem. Harry Lakser< 1) Proc. Univ. of Houston Lattice Theory Conf..Houston 1973 Disjointness conditions in free products of distributive lattices: An application of Ramsay's theorem. Harry Lakser< 1) 1. Introduction. Let L be

More information

The Arkhangel skiĭ Tall problem under Martin s Axiom

The Arkhangel skiĭ Tall problem under Martin s Axiom F U N D A M E N T A MATHEMATICAE 149 (1996) The Arkhangel skiĭ Tall problem under Martin s Axiom by Gary G r u e n h a g e and Piotr K o s z m i d e r (Auburn, Ala.) Abstract. We show that MA σ-centered

More information

(U) =, if 0 U, 1 U, (U) = X, if 0 U, and 1 U. (U) = E, if 0 U, but 1 U. (U) = X \ E if 0 U, but 1 U. n=1 A n, then A M.

(U) =, if 0 U, 1 U, (U) = X, if 0 U, and 1 U. (U) = E, if 0 U, but 1 U. (U) = X \ E if 0 U, but 1 U. n=1 A n, then A M. 1. Abstract Integration The main reference for this section is Rudin s Real and Complex Analysis. The purpose of developing an abstract theory of integration is to emphasize the difference between the

More information

VALUATION THEORY, GENERALIZED IFS ATTRACTORS AND FRACTALS

VALUATION THEORY, GENERALIZED IFS ATTRACTORS AND FRACTALS VALUATION THEORY, GENERALIZED IFS ATTRACTORS AND FRACTALS JAN DOBROWOLSKI AND FRANZ-VIKTOR KUHLMANN Abstract. Using valuation rings and valued fields as examples, we discuss in which ways the notions of

More information

MATH 13 SAMPLE FINAL EXAM SOLUTIONS

MATH 13 SAMPLE FINAL EXAM SOLUTIONS MATH 13 SAMPLE FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers.

More information

1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),

1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989), Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer

More information

Economics 204 Fall 2011 Problem Set 1 Suggested Solutions

Economics 204 Fall 2011 Problem Set 1 Suggested Solutions Economics 204 Fall 2011 Problem Set 1 Suggested Solutions 1. Suppose k is a positive integer. Use induction to prove the following two statements. (a) For all n N 0, the inequality (k 2 + n)! k 2n holds.

More information

Uniquely Universal Sets

Uniquely Universal Sets Uniquely Universal Sets 1 Uniquely Universal Sets Abstract 1 Arnold W. Miller We say that X Y satisfies the Uniquely Universal property (UU) iff there exists an open set U X Y such that for every open

More information

THE SET OF RECURRENT POINTS OF A CONTINUOUS SELF-MAP ON AN INTERVAL AND STRONG CHAOS

THE SET OF RECURRENT POINTS OF A CONTINUOUS SELF-MAP ON AN INTERVAL AND STRONG CHAOS J. Appl. Math. & Computing Vol. 4(2004), No. - 2, pp. 277-288 THE SET OF RECURRENT POINTS OF A CONTINUOUS SELF-MAP ON AN INTERVAL AND STRONG CHAOS LIDONG WANG, GONGFU LIAO, ZHENYAN CHU AND XIAODONG DUAN

More information

arxiv:math/ v2 [math.gn] 21 Jun 2002

arxiv:math/ v2 [math.gn] 21 Jun 2002 ON FINITE-DIMENSIONAL MAPS II arxiv:math/0202114v2 [math.gn] 21 Jun 2002 H. MURAT TUNCALI AND VESKO VALOV Abstract. Let f : X Y be a perfect n-dimensional surjective map of paracompact spaces and Y a C-space.

More information

MORE ABOUT SPACES WITH A SMALL DIAGONAL

MORE ABOUT SPACES WITH A SMALL DIAGONAL MORE ABOUT SPACES WITH A SMALL DIAGONAL ALAN DOW AND OLEG PAVLOV Abstract. Hušek defines a space X to have a small diagonal if each uncountable subset of X 2 disjoint from the diagonal, has an uncountable

More information

PM functions, their characteristic intervals and iterative roots

PM functions, their characteristic intervals and iterative roots ANNALES POLONICI MATHEMATICI LXV.2(1997) PM functions, their characteristic intervals and iterative roots by Weinian Zhang (Chengdu) Abstract. The concept of characteristic interval for piecewise monotone

More information

Problems - Section 17-2, 4, 6c, 9, 10, 13, 14; Section 18-1, 3, 4, 6, 8, 10; Section 19-1, 3, 5, 7, 8, 9;

Problems - Section 17-2, 4, 6c, 9, 10, 13, 14; Section 18-1, 3, 4, 6, 8, 10; Section 19-1, 3, 5, 7, 8, 9; Math 553 - Topology Todd Riggs Assignment 2 Sept 17, 2014 Problems - Section 17-2, 4, 6c, 9, 10, 13, 14; Section 18-1, 3, 4, 6, 8, 10; Section 19-1, 3, 5, 7, 8, 9; 17.2) Show that if A is closed in Y and

More information

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA  address: Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework

More information

On Riesz-Fischer sequences and lower frame bounds

On Riesz-Fischer sequences and lower frame bounds On Riesz-Fischer sequences and lower frame bounds P. Casazza, O. Christensen, S. Li, A. Lindner Abstract We investigate the consequences of the lower frame condition and the lower Riesz basis condition

More information

THE NON-URYSOHN NUMBER OF A TOPOLOGICAL SPACE

THE NON-URYSOHN NUMBER OF A TOPOLOGICAL SPACE THE NON-URYSOHN NUMBER OF A TOPOLOGICAL SPACE IVAN S. GOTCHEV Abstract. We call a nonempty subset A of a topological space X finitely non-urysohn if for every nonempty finite subset F of A and every family

More information

Q-LINEAR FUNCTIONS, FUNCTIONS WITH DENSE GRAPH, AND EVERYWHERE SURJECTIVITY

Q-LINEAR FUNCTIONS, FUNCTIONS WITH DENSE GRAPH, AND EVERYWHERE SURJECTIVITY MATH. SCAND. 102 (2008), 156 160 Q-LINEAR FUNCTIONS, FUNCTIONS WITH DENSE GRAPH, AND EVERYWHERE SURJECTIVITY F. J. GARCÍA-PACHECO, F. RAMBLA-BARRENO, J. B. SEOANE-SEPÚLVEDA Abstract Let L, S and D denote,

More information

A 3 3 DILATION COUNTEREXAMPLE

A 3 3 DILATION COUNTEREXAMPLE A 3 3 DILATION COUNTEREXAMPLE MAN DUEN CHOI AND KENNETH R DAVIDSON Dedicated to the memory of William B Arveson Abstract We define four 3 3 commuting contractions which do not dilate to commuting isometries

More information

Notes on ordinals and cardinals

Notes on ordinals and cardinals Notes on ordinals and cardinals Reed Solomon 1 Background Terminology We will use the following notation for the common number systems: N = {0, 1, 2,...} = the natural numbers Z = {..., 2, 1, 0, 1, 2,...}

More information

arxiv: v1 [math.lo] 10 Aug 2015

arxiv: v1 [math.lo] 10 Aug 2015 NAIVELY HAAR NULL SETS IN POLISH GROUPS arxiv:1508.02227v1 [math.lo] 10 Aug 2015 MÁRTON ELEKES AND ZOLTÁN VIDNYÁNSZKY Abstract. Let (G, ) be a Polish group. We say that a set X G is Haar null if there

More information

INTRODUCTION TO CARDINAL NUMBERS

INTRODUCTION TO CARDINAL NUMBERS INTRODUCTION TO CARDINAL NUMBERS TOM CUCHTA 1. Introduction This paper was written as a final project for the 2013 Summer Session of Mathematical Logic 1 at Missouri S&T. We intend to present a short discussion

More information

arxiv: v1 [math.gn] 2 Jul 2016

arxiv: v1 [math.gn] 2 Jul 2016 ON σ-countably TIGHT SPACES arxiv:1607.00517v1 [math.gn] 2 Jul 2016 ISTVÁN JUHÁSZ AND JAN VAN MILL Abstract. Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality

More information

Vitali sets and Hamel bases that are Marczewski measurable

Vitali sets and Hamel bases that are Marczewski measurable F U N D A M E N T A MATHEMATICAE 166 (2000) Vitali sets and Hamel bases that are Marczewski measurable by Arnold W. M i l l e r (Madison, WI) and Strashimir G. P o p v a s s i l e v (Sofia and Auburn,

More information

Quasi-invariant measures for continuous group actions

Quasi-invariant measures for continuous group actions Contemporary Mathematics Quasi-invariant measures for continuous group actions Alexander S. Kechris Dedicated to Simon Thomas on his 60th birthday Abstract. The class of ergodic, invariant probability

More information

ON THE LARGE TRANSFINITE INDUCTIVE DIMENSION OF A SPACE BY A NORMAL BASE. D. N. Georgiou, S. D. Iliadis, K. L. Kozlov. 1.

ON THE LARGE TRANSFINITE INDUCTIVE DIMENSION OF A SPACE BY A NORMAL BASE. D. N. Georgiou, S. D. Iliadis, K. L. Kozlov. 1. MATEMATIQKI VESNIK 61 (2009), 93 102 UDK 515.122 originalni nauqni rad research paper ON THE LARGE TRANSFINITE INDUCTIVE DIMENSION OF A SPACE BY A NORMAL BASE D. N. Georgiou, S. D. Iliadis, K. L. Kozlov

More information

ANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS Mathematics Subject Classification: 11B05, 11B13, 11P99

ANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS Mathematics Subject Classification: 11B05, 11B13, 11P99 ANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS N. HEGYVÁRI, F. HENNECART AND A. PLAGNE Abstract. We study the gaps in the sequence of sums of h pairwise distinct elements

More information

Axioms for Set Theory

Axioms for Set Theory Axioms for Set Theory The following is a subset of the Zermelo-Fraenkel axioms for set theory. In this setting, all objects are sets which are denoted by letters, e.g. x, y, X, Y. Equality is logical identity:

More information

MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017

MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 MATH 220 (all sections) Homework #12 not to be turned in posted Friday, November 24, 2017 Definition: A set A is finite if there exists a nonnegative integer c such that there exists a bijection from A

More information

van Rooij, Schikhof: A Second Course on Real Functions

van Rooij, Schikhof: A Second Course on Real Functions vanrooijschikhofproblems.tex December 5, 2017 http://thales.doa.fmph.uniba.sk/sleziak/texty/rozne/pozn/books/ van Rooij, Schikhof: A Second Course on Real Functions Some notes made when reading [vrs].

More information

UNIONS OF LINES IN F n

UNIONS OF LINES IN F n UNIONS OF LINES IN F n RICHARD OBERLIN Abstract. We show that if a collection of lines in a vector space over a finite field has dimension at least 2(d 1)+β, then its union has dimension at least d + β.

More information

Ultracompanions of subsets of a group

Ultracompanions of subsets of a group Comment.Math.Univ.Carolin. 55,2(2014) 257 265 257 Ultracompanions of subsets of a group I. Protasov, S. Slobodianiuk Abstract. LetGbeagroup, βgbethe Stone-Čech compactification ofgendowed with the structure

More information

arxiv: v2 [math.gn] 27 Sep 2010

arxiv: v2 [math.gn] 27 Sep 2010 THE GROUP OF ISOMETRIES OF A LOCALLY COMPACT METRIC SPACE WITH ONE END arxiv:0911.0154v2 [math.gn] 27 Sep 2010 ANTONIOS MANOUSSOS Abstract. In this note we study the dynamics of the natural evaluation

More information

MATH 3300 Test 1. Name: Student Id:

MATH 3300 Test 1. Name: Student Id: Name: Student Id: There are nine problems (check that you have 9 pages). Solutions are expected to be short. In the case of proofs, one or two short paragraphs should be the average length. Write your

More information

Countable subgroups of Euclidean space

Countable subgroups of Euclidean space Countable subgroups of Euclidean space Arnold W. Miller April 2013 revised May 21, 2013 In his paper [1], Konstantinos Beros proved a number of results about compactly generated subgroups of Polish groups.

More information

2 RENATA GRUNBERG A. PRADO AND FRANKLIN D. TALL 1 We thank the referee for a number of useful comments. We need the following result: Theorem 0.1. [2]

2 RENATA GRUNBERG A. PRADO AND FRANKLIN D. TALL 1 We thank the referee for a number of useful comments. We need the following result: Theorem 0.1. [2] CHARACTERIZING! 1 AND THE LONG LINE BY THEIR TOPOLOGICAL ELEMENTARY REFLECTIONS RENATA GRUNBERG A. PRADO AND FRANKLIN D. TALL 1 Abstract. Given a topological space hx; T i 2 M; an elementary submodel of

More information

GENERALIZED CANTOR SETS AND SETS OF SUMS OF CONVERGENT ALTERNATING SERIES

GENERALIZED CANTOR SETS AND SETS OF SUMS OF CONVERGENT ALTERNATING SERIES Journal of Applied Analysis Vol. 7, No. 1 (2001), pp. 131 150 GENERALIZED CANTOR SETS AND SETS OF SUMS OF CONVERGENT ALTERNATING SERIES M. DINDOŠ Received September 7, 2000 and, in revised form, February

More information

Projective Schemes with Degenerate General Hyperplane Section II

Projective Schemes with Degenerate General Hyperplane Section II Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 111-126. Projective Schemes with Degenerate General Hyperplane Section II E. Ballico N. Chiarli S. Greco

More information

MATH 13 FINAL EXAM SOLUTIONS

MATH 13 FINAL EXAM SOLUTIONS MATH 13 FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers. T F

More information

A product of γ-sets which is not Menger.

A product of γ-sets which is not Menger. A product of γ-sets which is not Menger. A. Miller Dec 2009 Theorem. Assume CH. Then there exists γ-sets A 0, A 1 2 ω such that A 0 A 1 is not Menger. We use perfect sets determined by Silver forcing (see

More information

A DECOMPOSITION THEOREM FOR FRAMES AND THE FEICHTINGER CONJECTURE

A DECOMPOSITION THEOREM FOR FRAMES AND THE FEICHTINGER CONJECTURE PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 A DECOMPOSITION THEOREM FOR FRAMES AND THE FEICHTINGER CONJECTURE PETER G. CASAZZA, GITTA KUTYNIOK,

More information

Free Subgroups of the Fundamental Group of the Hawaiian Earring

Free Subgroups of the Fundamental Group of the Hawaiian Earring Journal of Algebra 219, 598 605 (1999) Article ID jabr.1999.7912, available online at http://www.idealibrary.com on Free Subgroups of the Fundamental Group of the Hawaiian Earring Katsuya Eda School of

More information

ON THE MEASURABILITY OF FUNCTIONS WITH RESPECT TO CERTAIN CLASSES OF MEASURES

ON THE MEASURABILITY OF FUNCTIONS WITH RESPECT TO CERTAIN CLASSES OF MEASURES Georgian Mathematical Journal Volume 11 (2004), Number 3, 489 494 ON THE MEASURABILITY OF FUNCTIONS WITH RESPECT TO CERTAIN CLASSES OF MEASURES A. KHARAZISHVILI AND A. KIRTADZE Abstract. The concept of

More information

AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES

AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov

More information

Homotopy and homology groups of the n-dimensional Hawaiian earring

Homotopy and homology groups of the n-dimensional Hawaiian earring F U N D A M E N T A MATHEMATICAE 165 (2000) Homotopy and homology groups of the n-dimensional Hawaiian earring by Katsuya E d a (Tokyo) and Kazuhiro K a w a m u r a (Tsukuba) Abstract. For the n-dimensional

More information

Yuqing Chen, Yeol Je Cho, and Li Yang

Yuqing Chen, Yeol Je Cho, and Li Yang Bull. Korean Math. Soc. 39 (2002), No. 4, pp. 535 541 NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY Yuqing Chen, Yeol Je Cho, and Li Yang Abstract. In this paper, we introduce the concept of lower semicontinuous

More information

On an algebraic version of Tamano s theorem

On an algebraic version of Tamano s theorem @ Applied General Topology c Universidad Politécnica de Valencia Volume 10, No. 2, 2009 pp. 221-225 On an algebraic version of Tamano s theorem Raushan Z. Buzyakova Abstract. Let X be a non-paracompact

More information

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 27

Discrete Mathematics for CS Spring 2007 Luca Trevisan Lecture 27 CS 70 Discrete Mathematics for CS Spring 007 Luca Trevisan Lecture 7 Infinity and Countability Consider a function f that maps elements of a set A (called the domain of f ) to elements of set B (called

More information

S. Mrówka introduced a topological space ψ whose underlying set is the. natural numbers together with an infinite maximal almost disjoint family(madf)

S. Mrówka introduced a topological space ψ whose underlying set is the. natural numbers together with an infinite maximal almost disjoint family(madf) PAYNE, CATHERINE ANN, M.A. On ψ (κ, M) spaces with κ = ω 1. (2010) Directed by Dr. Jerry Vaughan. 30pp. S. Mrówka introduced a topological space ψ whose underlying set is the natural numbers together with

More information