Bucket handles and Solenoids Notes by Carl Eberhart, March 2004
|
|
- Jocelin Bennett
- 5 years ago
- Views:
Transcription
1 Bucket handles and Solenoids Notes by Carl Eberhart, March Introduction A continuum is a nonempty, compact, connected metric space. A nonempty compact connected subspace of a continuum X is called a subcontinuum of X. A useful theorem about subcontinua, hich every first year topology student should prove is: 1.1. Theorem. The intersection of a toer of subcontinua of X is a subcontinuum of X. A continuum is indecomposable if it cannot be ritten as the union of to proper subcontinua. The composant of a point x in a continuum X is the union of all proper subcontinua of X hich contain x. Here is a nice theorem. 1.. Theorem. The composants of a continuum are dense. The composants of an indecomposable continuum are pairise disjoint. A classical example by contstructed by B. Knaster in the early 190 s is still of interest The Bucket handle continuum. Let B 0 be the set of all closed semicircles in the upper half plane centered on c 0 =.5 hose diameters have enpoints in the Cantor middle third set. Let (B 0 ) denote the reflection of B 0 about the x-axis. For 1 n, let B n = c n c 0 + B 0 3 n, here c n =.5 3 n. Then K = B 0 B 1 B is an indecomposable continuum. The union of the semicircles hose endpoints are endpoints of the Cantor set is called the visible composant. In the figure belo, a portion of the visible composant is shon. It is a union of a toer of arcs all ith (0,0) as one endpoint. (An arc is a continuum hich is homeomorphic ith the unit interval I = [0,1].)
2 It is possible to see the bucket handle as an intersection of a toer of -cells, each one obtained from the preceeding one by digging a canal out of it. Why is K indecomposable? Well, if you can convince yourself that K (1) has only arcs for proper subcontinua, and () has no arcs ith interior (in K ), then it is pretty easy to argue that K is indecomposable, for otherise it ould be the union of to arcs A and B and A - B ould be a nonempty subset of the interior of K. On the other hand there is another ay to construct indecomposable continua hich makes it possible to prove indecomposability rather easily.. The inverse limit construction The inverse limit X = lim (X i, f i ) of a sequence (X i ) 1 of continua and surjective maps f i : X i+1 X i is defined as the intersection of the subsets Q n = {(x i ) 1 such that x i = f i (x i+1 ) for all i = 1,, n} of the product Π i=1 X i. The spaces X i are called the factor spaces of X ; the maps f i are called the bonding maps of the inverse limit. For each positive integer i, the map π i : X X i given by π i ((x i ) 1 ) = x i is called the i th projection map. It is continuous since it is the restriction of the projection ρ i : Π n=1 X n X i to X..1. Theorem. The inverse limit X of continua is a continuum. Further if A is a subcontinuum of X then A = lim (A i, g i ), here A i = π i (A) and g i = f i Ai+1. Proof. X = n=1 Q n is a continuum by 1.1, since for each n, Q n is homeomorphic ith Π i=n. Further, (a i) 1 A if and only for each i, a i A i and g i+1 (a i+1 ) = f i+1 (a i+1 ) = a i. One easy ay to decide if to inverse limits are homeomorphic is described in the next theorem... Theorem. To inverse limits X = lim (X i, f i ) and Y = lim (Y i, g i ) are homeomorphic if there is a sequence of homeomorphisms h i : X i Y i such that h i f i (x) = g i h i+1 (x) for each i and all x X i+1. Proof. Define a function h : X Y by h ((x i ) i=1 ) = (h i(x i )) i=1 for each positive integer i. h is into Y because g i h i+1 (x i+1 ) = h i f i (x i+1 ) = h i (x i ), for each i. h is continuous since π i h = h i π i is continuous for each i. Call h the map induced by the commutative diagram. f 1 f X 1 X Xi X h 1 h h i h g 1 g Y 1 Y Yi f i g i Y In the same ay, using the inverse homeomorphisms h 1 i : Y i X i, e can define an induced map from Y to X and sho that it is the inverse of h. Thus a map induced by homeomorphisms is a homeomorphis.
3 Here is the theorem describing the indecomposability of a continuum constructed as an inverse limit. A map f : X Y is called indecomposable provided henever X = A B, then f(a) = Y or f(b) = Y. So a continuum is indecomposable if its identity map is indecomposable..3. Theorem. X is indecomposable provided each of its bonding maps is indecomposable. Proof. Suppose X is the union of to proper subcontinua A and B. Then there is an n 1 so that π n1 (A) X n1, otherise by.1, X = A. Note also that for all n n 1, π n (A) X n. In the same ay, there is an n so that π n (B) X n. So if e let n be the maximum of n 1 and n, then π n (A) and π n (B) are proper subcontinua of X n. Hoever, π n+1 (X ) = X n+1 = π n+1 (A) π n+1 (B), f n (π n+1 (A)) = π n (A), and f n (π n+1 (B)) = π n (B), a contradiction to the assumed property of the bonding maps f i. Another classical indecomposable continuum, studied somehat later [] than the bucket handle is the dyadic solenoid S = lim (X i, f i ), here each factor space X i is the unit circle S 1 and each bonding map f i is the squaring map z..4. Corollary. S is indecomposable. Proof. The squaring map is indecomposable, so the theorem follos from.3. Note that any inverse limit of circles here the bonding maps are poer maps z i, i is indecomposable by the same argument. Such continua are referred to as solenoids. Solenoids are the only indecomposable continua hich admit a group mulitiplication[]. Another class of indecomposable continua are the Knaster continua, hich are defined as the continua obtained as inverse limits by using the unit interval I as the factor space and standard maps n, n > 1, defined by { nx i if i is even and 0 i n (x) = n x i + n 1 1, i + 1 nx if i is odd and 0 < n i x i + n 1 1 Clearly, the map n : I I is indecomposable hen n > 1, and so one has the corollary..5. Corollary. Any Knaster continuum is indecomposable..6. Exercise. Sho that the dyadic solenoid has the property that each proper subcontinuum is an arc, and has a basis of neighborhoods consisting of sets homeomorphic ith the Cantor set crossed ith an arc An embedding theorem A homeomorphism of a space X onto a subspace of a space Y is called an embedding of X into Y Theorem. (A version of the Anderson-Choquet embedding theorem) Suppose the factor spaces X i of X are all embedded in a common continuum X (ith metric d, say) and the bonding maps f i satisfy the conditions: (1) There is a positive number K so that
4 4 for each i and each x X i+1, d(x, f i (x)) < K i, and () for each factor space X i and each positive δ, there is a positive δ so that if k > i and p, q X k ith d(f ik (p), f ik (q)) > δ then d(p, q) > δ. Then for each x = (x i ) i X, the sequnce of coordinates x i converges to a unique point h(x) X and the function h : X X is an embedding. Proof. Condition (1) suffices to guarantee that the sequence of coordinates of a point in X is Cauchy, and so converges. So the function h is ell defined. To see that h is continuous, let ɛ > 0 be given. Choose N so large that K i=n i < ɛ/4 and and hence by (1) d(x N, h(x)) d(x N, x N+1 ) + d(x N+1, h(x)) K/ N + d(x N+1, h(x)) < ɛ/4 for all x X. Hence if x X and U is an open set in X N about x N of diameter less than ɛ/4. Then π 1 N (U) is a basic open set in X. Further, if y = (y i ) i π 1 N (U), then y N U, and so d(h(x), h(y) d(h(x), x N ) + d(x N, y N ) + d(y N, h(y)) < ɛ. This shos h is continuous. To see that h is 1-1, suppose x y here x and y are in X. Since x y, there is an i 0 so that x i0 y i0. Let δ = d(x i0, y i0 )/, and use () to get a δ \prime > 0 so that for k > i 0 and p, q X k, if d(f i0 k(p), f i0 k(q)) > δ, then d(p, q) > δ. Hence for k > i 0, d(x k, y k ) > δ, and so d(h(x), h(y)) δ. This shos h is an embedding. 3.. Theorem. K is homeomorphic ith an inverse limit of arcs ith indecomposable bonding maps. Proof. To apply 3.1 to K, for each positive integer i, let a i = ( i 6, i 6 ), the bottom of the (i + 1) st set of semicircles B i counting from the right, and let A i be the subarc of the visible composant ith endpoints a 0 and a i. These arcs are the factor spaces. The bonding map r i : A i+1 A i is the retraction hich projects a point on the first quarter circle starting at a i+1 horizontally to the right onto the matching point on the end quartercircle of A i, and otherise projects a point radially onto the nearest point of A i, except near a i+1. There the last quarter circle of A i+1 is mapped homemorphically onto the circular arc from a 0 to (.5 + cos((1 (1/) i )π),.5 + sin((1 (1/) i )π)), and the circular arc from (1 (1/) i )π to π on the last semicircle of A i+1 is mapped homeomorphically onto the circular arc from (.5 + cos((1 (1/) i )π),.5 + sin((1 (1/) i )π)) to (.5 + cos((1 (1/) i )π),.5 + \sin((1 (1/) i )π)). It is easy to see that r i is indecomposable. Note that d(x, r 1 (x)) < 1/3, d(x, r (x)) < 1/3, and in general d(x, r i (x)) < 1/3 i, so condition (1) of.3 is satisfied ith K = 1. So the function h : X K is ell-defined and continuous. Since K = i=1 A i, the continuity of h guarantees that h is onto. We verify that h is 1-1 directly. Suppose h(x) = h(y). Then for some n, x n and y n lie in the same circular arc of A n. Then from then on the coordinates x n+i and y n+i lie along the outer third of the radii dran from the center of that circular arc. But since h(x) = h(y), for some m > n, x i and y i lie in the same radius for all i > m. But then x i = r i (x i+1 ) = r i (y i+1 ) = y i for all i > m. So x = y, and h is 1-1. So e have realized K as the inverse limit lim (A i, r i ) of arcs ith indecomposable bonding maps. No e ant to prove the 3.3. Theorem. K is homeomorphic ith the Knaster continuum lim (I, ).
5 5 Proof. Let g 1 be any homeomorphism from A 1 to I such that g 1 (a 0 ) = 0 and g 1 (a 1 ) = 1. Then define g : A I by g 1 (x) g (x) = if x A 1 1 g 1(r 1 (x)) if x A A 1 The function g is a homeomorphism such that g 1 r 1 = g. Using the same method, e successively define homeomorphisms g i : A i I so that g i 1 r i 1 = g i. This sequence induces a homeomorphism g : lim (A i, r i ) lim (I, ). r A 1 r 1 A Ai r i lim (A i, r i ) g 1 g g i g I I I lim (I, ) 4. Where is (x i ) i=1? It is an interesting exercise to ask here a given point in lim (I, ) goes under the homeomorphism h(h ) 1 : lim (I, ) K constructed previously. For example, it is easy to see hat point goes to a 0 = (0, 0): the point (0, 0, ). Where does the point x = (1, 1/, 1/4, )go? First find its image y = (h ) 1 (x) in lim (A i, r i ), then push that on to h(y) in K. By the ay h as constructed, e can compute that y = (a 1, a 1, a 1, ). Then h(y) = a 1. Every inverse limit ith one factor space and one bonding map has a shift homeomorphism, s, defined by s((x 1, x, x 3, )) = (x, x 3, ). So the bucket handle K has a shift homeomorphism, s = h(h ) 1 sh h 1 : K K. Where does the shift homeomorphism take the point a 0? It is pretty easy to see that this point is fixed under the shift. What about s(a 1 )? First push a 1 to (1, 1/, 1/4, ), then shift it to (1/, 1/4, ), then pull it back to (h 1 1 (1/), h 1 1 (1/), h 1 1 (1/), ) in lim (A i, r i ) then push to h 1 1 (1/) under h. Note that the shift homeomorphism on K depends on the homemorphism h 1 of I to A 1. Hoever, the fixed point a 0 of s does not depend on h Exercise: There is another fixed point of the shift s defined above. Where does it come from in lim (I, ) and here is it at in K? Does its location in K depend on h 1? 5. Mapping the dyadic solenoid lightly onto the bucket handle David Bellamy [1] as the first to see this, I think. First you have to think of the group S 1 as the quotient space I/{0 = 1} of I ith the operation of addition mod { 1. In this setting, the squaring map on S 1 becomes the doubling t if 0 t 1/ mod 1 map on I. dbl(t) = t 1 if 1/ t 1
6 6 Note that this is continuous on I/{0 = 1}, and commutes ith. Hence, there is an induced map of the solenoid onto the bucket handle. dbl dbl I/{0 = 1} I/{0 = 1} S I I lim (I, ) 5.1. Exercise. Use the diagram above to sho that S /{x = x} = K. 6. The hyperspaces of subcontinua of K and S If X is any continuum ith metric d say, and A and B are subcontinua of X, the Hausdorff distance beteen A and B, H d (A, B) is defined by H d (A, B) = inf{\epsilon each point of either set is ithin ɛ of some point of the other.} The set of all subcontinua of X is denoted by C(X) Theorem. H d is a metric on C(X), and C(X) topologized by this metric is a continuum. C(X), topologized ith the Hausdorff metric, is called the hyperspace of subcontinua of X 6.. Theorem. C(S ) is homeomorphic ith the cone over S. C(K ) is homeomorphic ith the cone over K. References [1] David Bellamy, A tree-like continuum ithout the fixed point property, Houston Math. J. 6 (1979), [] Anthonie van Heemert, Topologische Gruppen und unzerlegbar Kontinua, Compositio Math. 5 (1937), [3] Sam B. Nadler, Jr. Continuum Theory. Marcel Dekker, 199. [4] K. Kuratoski, Topology, vol. 1, Amsterdam, 1968.
INDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS
INDECOMPOSABILITY IN INVERSE LIMITS WITH SET-VALUED FUNCTIONS JAMES P. KELLY AND JONATHAN MEDDAUGH Abstract. In this paper, we develop a sufficient condition for the inverse limit of upper semi-continuous
More informationTHE OPENNESS OF INDUCED MAPS ON HYPERSPACES
C O L L O Q U I U M M A T H E M A T I C U M VOL. 74 1997 NO. 2 THE OPENNESS OF INDUCED MAPS ON HYPERSPACES BY ALEJANDRO I L L A N E S (MÉXICO) A continuum is a compact connected metric space. A map is
More informationMAPPING CHAINABLE CONTINUA ONTO DENDROIDS
MAPPING CHAINABLE CONTINUA ONTO DENDROIDS PIOTR MINC Abstract. We prove that every chainable continuum can be mapped into a dendroid such that all point-inverses consist of at most three points. In particular,
More informationTOPOLOGICAL ENTROPY AND TOPOLOGICAL STRUCTURES OF CONTINUA
TOPOLOGICAL ENTROPY AND TOPOLOGICAL STRUCTURES OF CONTINUA HISAO KATO, INSTITUTE OF MATHEMATICS, UNIVERSITY OF TSUKUBA 1. Introduction During the last thirty years or so, many interesting connections between
More informationTopology 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 informationMTG 5316/4302 FALL 2018 REVIEW FINAL
MTG 5316/4302 FALL 2018 REVIEW FINAL JAMES KEESLING Problem 1. Define open set in a metric space X. Define what it means for a set A X to be connected in a metric space X. Problem 2. Show that if a set
More informationDECOMPOSITIONS OF HOMOGENEOUS CONTINUA
PACIFIC JOURNAL OF MATHEMATICS Vol. 99, No. 1, 1982 DECOMPOSITIONS OF HOMOGENEOUS CONTINUA JAMES T. ROGERS, JR. The purpose of this paper is to present a general theory of decomposition of homogeneous
More informationRetractions and contractibility in hyperspaces
Topology and its Applications 54 (007) 333 338 www.elsevier.com/locate/topol Retractions and contractibility in hyperspaces Janusz J. Charatonik a,b, Patricia Pellicer-Covarrubias c, a Instituto de Matemáticas,
More informationKELLEY REMAINDERS OF [0, ) ROBBIE A. BEANE AND W LODZIMIERZ J. CHARATONIK
TOPOLOGY PROCEEDINGS Volume 32, No. 1, 2008 Pages 1-14 http://topology.auburn.edu/tp/ KELLEY REMAINDERS OF [0, ) ROBBIE A. BEANE AND W LODZIMIERZ J. CHARATONIK Abstract. We investigate Kelley continua
More informationTopology 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: 046-424
More informationSingularities of meager composants and filament composants
Singularities of meager composants and filament composants David Sumner Lipham Department of Mathematics, Auburn University, Auburn, AL 36849 Abstract arxiv:1806.10828v11 [math.gn] 21 Oct 2018 Suppose
More informationINVERSE LIMITS WITH SET VALUED FUNCTIONS. 1. Introduction
INVERSE LIMITS WITH SET VALUED FUNCTIONS VAN NALL Abstract. We begin to answer the question of which continua in the Hilbert cube can be the inverse limit with a single upper semi-continuous bonding map
More informationHouston Journal of Mathematics. c 2004 University of Houston Volume 30, No. 4, 2004
Houston Journal of Mathematics c 2004 University of Houston Volume 30, No. 4, 2004 ATRIODIC ABSOLUTE RETRACTS FOR HEREDITARILY UNICOHERENT CONTINUA JANUSZ J. CHARATONIK, W LODZIMIERZ J. CHARATONIK AND
More informationSOME STRUCTURE THEOREMS FOR INVERSE LIMITS WITH SET-VALUED FUNCTIONS
http://topology.auburn.edu/tp/ TOPOLOGY PROCEEDINGS Volume 42 (2013) Pages 237-258 E-Published on January 10, 2013 SOME STRUCTURE THEOREMS FOR INVERSE LIMITS WITH SET-VALUED FUNCTIONS M. M. MARSH Abstract.
More informationBloom Filters and Locality-Sensitive Hashing
Randomized Algorithms, Summer 2016 Bloom Filters and Locality-Sensitive Hashing Instructor: Thomas Kesselheim and Kurt Mehlhorn 1 Notation Lecture 4 (6 pages) When e talk about the probability of an event,
More informationarxiv: v1 [math.gn] 29 Aug 2016
A FIXED-POINT-FREE MAP OF A TREE-LIKE CONTINUUM INDUCED BY BOUNDED VALENCE MAPS ON TREES RODRIGO HERNÁNDEZ-GUTIÉRREZ AND L. C. HOEHN arxiv:1608.08094v1 [math.gn] 29 Aug 2016 Abstract. Towards attaining
More informationTopology 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 informationIntroduction to Topology
Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about
More informationTHE HYPERSPACE C 2 (X) FOR A FINITE GRAPH X IS UNIQUE. Alejandro Illanes
GLASNIK MATEMATIČKI Vol. 37(57)(2002), 347 363 THE HYPERSPACE C 2 (X) FOR A FINITE GRAPH X IS UNIQUE Alejandro Illanes Universidad Nacional Autónoma de México, México Abstract. Let X be a metric continuum.
More informationStat 8112 Lecture Notes Weak Convergence in Metric Spaces Charles J. Geyer January 23, Metric Spaces
Stat 8112 Lecture Notes Weak Convergence in Metric Spaces Charles J. Geyer January 23, 2013 1 Metric Spaces Let X be an arbitrary set. A function d : X X R is called a metric if it satisfies the folloing
More informationTopology 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 informationResearch Announcement: ARCWISE CONNECTED CONTINUA AND THE FIXED POINT PROPERTY
Volume 1, 1976 Pages 345 349 http://topology.auburn.edu/tp/ Research Announcement: ARCWISE CONNECTED CONTINUA AND THE FIXED POINT PROPERTY by J. B. Fugate and L. Mohler Topology Proceedings Web: http://topology.auburn.edu/tp/
More informationTerminal continua and quasi-monotone mappings
Topology and its Applications 47 (1992) 69-77 North-Holland 69 Terminal continua and quasi-monotone mappings J.J. Charatonik Mathematical Institute, University of Wroclaw, pl. Grunwaldzki 2/4, 50-384 Wroclaw,
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More information7 Complete metric spaces and function spaces
7 Complete metric spaces and function spaces 7.1 Completeness Let (X, d) be a metric space. Definition 7.1. A sequence (x n ) n N in X is a Cauchy sequence if for any ɛ > 0, there is N N such that n, m
More informationA homogeneous continuum without the property of Kelley
Topology and its Applications 96 (1999) 209 216 A homogeneous continuum without the property of Kelley Włodzimierz J. Charatonik a,b,1 a Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4,
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationMetric Spaces Math 413 Honors Project
Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only
More information1. Simplify the following. Solution: = {0} Hint: glossary: there is for all : such that & and
Topology MT434P Problems/Homework Recommended Reading: Munkres, J.R. Topology Hatcher, A. Algebraic Topology, http://www.math.cornell.edu/ hatcher/at/atpage.html For those who have a lot of outstanding
More informationApplications of Homotopy
Chapter 9 Applications of Homotopy In Section 8.2 we showed that the fundamental group can be used to show that two spaces are not homeomorphic. In this chapter we exhibit other uses of the fundamental
More informationA GENERALIZATION OF KELLEY S THEOREM FOR C-SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 5, Pages 1537 1541 S 0002-9939(99)05158-8 Article electronically published on October 5, 1999 A GENERALIZATION OF KELLEY S THEOREM FOR
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More informationCOLLOQUIUM MA THEMA TICUM
COLLOQUIUM MA THEMA TICUM VOL. 75 1998 NO.2 HEREDITARILY WEAKLY CONFLUENT INDUCED MAPPINGS ARE HOMEOMORPHISMS BY JANUSZ J. CHARATONIK AND WLODZIMIERZ J. CHARATONIK (WROCLAW AND MEXICO, D.F.) For a given
More informationHUSSAM ABOBAKER AND W LODZIMIERZ J. CHARATONIK
T -CLOSED SETS HUSSAM ABOBAKER AND W LODZIMIERZ J. CHARATONIK Abstract. A subset A of a continuum X is called T -closed set if T (A) = A, where T denotes the Jones T - function. We give a characterization
More informationMH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then
MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever
More information1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More informationTREE-LIKE CONTINUA THAT ADMIT POSITIVE ENTROPY HOMEOMORPHISMS ARE NON-SUSLINEAN
TREE-LIKE CONTINUA THAT ADMIT POSITIVE ENTROPY HOMEOMORPHISMS ARE NON-SUSLINEAN CHRISTOPHER MOURON Abstract. t is shown that if X is a tree-like continuum and h : X X is a homeomorphism such that the topological
More informationSMOOTHNESS OF HYPERSPACES AND OF CARTESIAN PRODUCTS
SMOOTHNESS OF HYPERSPACES AND OF CARTESIAN PRODUCTS W LODZIMIERZ J. CHARATONIK AND W LADYS LAW MAKUCHOWSKI Abstract. We show that for any continua X and Y the smoothness of either the hyperspace C(X) or
More informationSolve EACH of the exercises 1-3
Topology Ph.D. Entrance Exam, August 2011 Write a solution of each exercise on a separate page. Solve EACH of the exercises 1-3 Ex. 1. Let X and Y be Hausdorff topological spaces and let f: X Y be continuous.
More informationQ & A in General Topology, Vol. 16 (1998)
Q & A in General Topology, Vol. 16 (1998) QUESTIONS ON INDUCED UNIVERSAL MAPPINGS JANUSZ J. CHARATONIK AND WLODZIMIERZ J. CHARATONIK University of Wroclaw (Wroclaw, Poland) Universidad Nacional Aut6noma
More informationTopology 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 informationMAPPINGS BETWEEN INVERSE LIMITS OF CONTINUA WITH MULTIVALUED BONDING FUNCTIONS
MAPPINGS BETWEEN INVERSE LIMITS OF CONTINUA WITH MULTIVALUED BONDING FUNCTIONS W LODZIMIERZ J. CHARATONIK AND ROBERT P. ROE Abstract. We investigate the limit mappings between inverse limits of continua
More informationTOPOLOGY HW 2. x x ± y
TOPOLOGY HW 2 CLAY SHONKWILER 20.9 Show that the euclidean metric d on R n is a metric, as follows: If x, y R n and c R, define x + y = (x 1 + y 1,..., x n + y n ), cx = (cx 1,..., cx n ), x y = x 1 y
More informationMath General Topology Fall 2012 Homework 8 Solutions
Math 535 - General Topology Fall 2012 Homework 8 Solutions Problem 1. (Willard Exercise 19B.1) Show that the one-point compactification of R n is homeomorphic to the n-dimensional sphere S n. Note that
More informationTopology 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 informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationMath 426 Homework 4 Due 3 November 2017
Math 46 Homework 4 Due 3 November 017 1. Given a metric space X,d) and two subsets A,B, we define the distance between them, dista,b), as the infimum inf a A, b B da,b). a) Prove that if A is compact and
More informationADDING MACHINES, ENDPOINTS, AND INVERSE LIMIT SPACES. 1. Introduction
ADDING MACHINES, ENDPOINTS, AND INVERSE LIMIT SPACES LORI ALVIN AND KAREN BRUCKS Abstract. Let f be a unimodal map in the logistic or symmetric tent family whose restriction to the omega limit set of the
More informationThe Fundamental Group and Covering Spaces
Chapter 8 The Fundamental Group and Covering Spaces In the first seven chapters we have dealt with point-set topology. This chapter provides an introduction to algebraic topology. Algebraic topology may
More informationMASTERS EXAMINATION IN MATHEMATICS SOLUTIONS
MASTERS EXAMINATION IN MATHEMATICS PURE MATHEMATICS OPTION SPRING 010 SOLUTIONS Algebra A1. Let F be a finite field. Prove that F [x] contains infinitely many prime ideals. Solution: The ring F [x] of
More informationFrom continua to R trees
1759 1784 1759 arxiv version: fonts, pagination and layout may vary from AGT published version From continua to R trees PANOS PAPASOGLU ERIC SWENSON We show how to associate an R tree to the set of cut
More informationClassification of matchbox manifolds
Classification of matchbox manifolds Steven Hurder University of Illinois at Chicago www.math.uic.edu/ hurder Workshop on Dynamics of Foliations Steven Hurder (UIC) Matchbox Manifolds April 28, 2010 1
More information4 Countability axioms
4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said
More informationMath General Topology Fall 2012 Homework 6 Solutions
Math 535 - General Topology Fall 202 Homework 6 Solutions Problem. Let F be the field R or C of real or complex numbers. Let n and denote by F[x, x 2,..., x n ] the set of all polynomials in n variables
More informationDendrites, Topological Graphs, and 2-Dominance
Marquette University e-publications@marquette Mathematics, Statistics and Computer Science Faculty Research and Publications Mathematics, Statistics and Computer Science, Department of 1-1-2009 Dendrites,
More informationCOMPLEXITY OF SETS AND BINARY RELATIONS IN CONTINUUM THEORY: A SURVEY
COMPLEXITY OF SETS AND BINARY RELATIONS IN CONTINUUM THEORY: A SURVEY ALBERTO MARCONE Contents 1. Descriptive set theory 2 1.1. Spaces of continua 2 1.2. Descriptive set theoretic hierarchies 3 1.3. Descriptive
More informationImmerse Metric Space Homework
Immerse Metric Space Homework (Exercises -2). In R n, define d(x, y) = x y +... + x n y n. Show that d is a metric that induces the usual topology. Sketch the basis elements when n = 2. Solution: Steps
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationDIFFERENTIAL GEOMETRY 1 PROBLEM SET 1 SOLUTIONS
DIFFERENTIAL GEOMETRY PROBLEM SET SOLUTIONS Lee: -4,--5,-6,-7 Problem -4: If k is an integer between 0 and min m, n, show that the set of m n matrices whose rank is at least k is an open submanifold of
More informationSHADOWING AND ω-limit SETS OF CIRCULAR JULIA SETS
SHADOWING AND ω-limit SETS OF CIRCULAR JULIA SETS ANDREW D. BARWELL, JONATHAN MEDDAUGH, AND BRIAN E. RAINES Abstract. In this paper we consider quadratic polynomials on the complex plane f c(z) = z 2 +
More informationHOMOGENEITY DEGREE OF SOME SYMMETRIC PRODUCTS
HOMOGENEITY DEGREE OF SOME SYMMETRIC PRODUCTS RODRIGO HERNÁNDEZ-GUTIÉRREZ AND VERÓNICA MARTÍNEZ-DE-LA-VEGA Abstract. For a metric continuum X, we consider the n th -symmetric product F n(x) defined as
More informationHOMOGENEOUS CIRCLE-LIKE CONTINUA THAT CONTAIN PSEUDO-ARCS
Volume 1, 1976 Pages 29 32 http://topology.auburn.edu/tp/ HOMOGENEOUS CIRCLE-LIKE CONTINUA THAT CONTAIN PSEUDO-ARCS by Charles L. Hagopian Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail:
More informationANALYSIS WORKSHEET II: METRIC SPACES
ANALYSIS WORKSHEET II: METRIC SPACES Definition 1. A metric space (X, d) is a space X of objects (called points), together with a distance function or metric d : X X [0, ), which associates to each pair
More informationRepresentation space with confluent mappings
Representation space with confluent mappings José G. Anaya a, Félix Capulín a, Enrique Castañeda-Alvarado a, W lodzimierz J. Charatonik b,, Fernando Orozco-Zitli a a Universidad Autónoma del Estado de
More informationIntroduction to Dynamical Systems
Introduction to Dynamical Systems France-Kosovo Undergraduate Research School of Mathematics March 2017 This introduction to dynamical systems was a course given at the march 2017 edition of the France
More informationUNIVERSALITY OF WEAKLY ARC-PRESERVING MAPPINGS
UNIVERSALITY OF WEAKLY ARC-PRESERVING MAPPINGS Janusz J. Charatonik and W lodzimierz J. Charatonik Abstract We investigate relationships between confluent, semiconfluent, weakly confluent, weakly arc-preserving
More informationCONTINUUM-CHAINABLE CONTINUUM WHICH CAN NOT BE MAPPED ONTO AN ARCWISE CONNECTED CONTINUUM BY A MONOTONE EPSILON MAPPING
GLASNIK MATEMATIČKI Vol. 48(68)(2013), 167 172 CONTINUUM-CHAINABLE CONTINUUM WHICH CAN NOT BE MAPPED ONTO AN ARCWISE CONNECTED CONTINUUM BY A MONOTONE EPSILON MAPPING Pavel Pyrih, Benjamin Vejnar and Luis
More informationMATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1
MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION ELEMENTS OF SOLUTION Problem 1 1. Let X be a Hausdorff space and K 1, K 2 disjoint compact subsets of X. Prove that there exist disjoint open sets U 1 and
More informationAfter taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain.
Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, the motivating insight behind topology is that some geometric
More informationProperties of Nonmetric Hereditarily Indecomposable Subcontinua of. Finite Products of Lexicographic Arcs
Properties of Nonmetric Hereditarily Indecomposable Subcontinua of Finite Products of Lexicographic Arcs Except where reference is made to the work of others, the work described in this dissertation is
More informationProblem Set 5. 2 n k. Then a nk (x) = 1+( 1)k
Problem Set 5 1. (Folland 2.43) For x [, 1), let 1 a n (x)2 n (a n (x) = or 1) be the base-2 expansion of x. (If x is a dyadic rational, choose the expansion such that a n (x) = for large n.) Then the
More informationSolutions to Tutorial 8 (Week 9)
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 8 (Week 9) MATH3961: Metric Spaces (Advanced) Semester 1, 2018 Web Page: http://www.maths.usyd.edu.au/u/ug/sm/math3961/
More informationA Core Decomposition of Compact Sets in the Plane
A Core Decomposition of Compact Sets in the Plane Benoît Loridant 1, Jun Luo 2, and Yi Yang 3 1 Montanuniversität Leoben, Franz Josefstrasse 18, Leoben 8700, Austria 2,3 School of Mathematics, Sun Yat-Sen
More informationMath 320-2: Midterm 2 Practice Solutions Northwestern University, Winter 2015
Math 30-: Midterm Practice Solutions Northwestern University, Winter 015 1. Give an example of each of the following. No justification is needed. (a) A metric on R with respect to which R is bounded. (b)
More informationInverse Limit Spaces
Inverse Limit Spaces Except where reference is made to the work of others, the work described in this thesis is my own or was done in collaboration with my advisory committee. This thesis does not include
More informationMetric Spaces Math 413 Honors Project
Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only
More informationHomework 4: Mayer-Vietoris Sequence and CW complexes
Homework 4: Mayer-Vietoris Sequence and CW complexes Due date: Friday, October 4th. 0. Goals and Prerequisites The goal of this homework assignment is to begin using the Mayer-Vietoris sequence and cellular
More informationNAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key
NAME: Mathematics 205A, Fall 2008, Final Examination Answer Key 1 1. [25 points] Let X be a set with 2 or more elements. Show that there are topologies U and V on X such that the identity map J : (X, U)
More informationINVERSE FUNCTION THEOREM and SURFACES IN R n
INVERSE FUNCTION THEOREM and SURFACES IN R n Let f C k (U; R n ), with U R n open. Assume df(a) GL(R n ), where a U. The Inverse Function Theorem says there is an open neighborhood V U of a in R n so that
More informationTopology. 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 informationCHAPTER 7. Connectedness
CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set
More informationIntroduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION
Introduction to Proofs in Analysis updated December 5, 2016 By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Purpose. These notes intend to introduce four main notions from
More informationThe Property of Kelley and Confluent Mappings
BULLETIN OF nte POLISH ACADEMY OF SCIENCES Mathematics Y0l. 31 No. 9--12, 1983 GENERAL TOPOLOGY The Property of Kelley and Confluent Mappings by Janusz J. CHARA TONIK Presented by K. URBANIK on April 9,
More information(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define
Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that
More informationTOPOLOGY TAKE-HOME CLAY SHONKWILER
TOPOLOGY TAKE-HOME CLAY SHONKWILER 1. The Discrete Topology Let Y = {0, 1} have the discrete topology. Show that for any topological space X the following are equivalent. (a) X has the discrete topology.
More informationAlgebraic Topology M3P solutions 1
Algebraic Topology M3P21 2015 solutions 1 AC Imperial College London a.corti@imperial.ac.uk 9 th February 2015 (1) (a) Quotient maps are continuous, so preimages of closed sets are closed (preimages of
More informationTopology 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 informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationMATH 8253 ALGEBRAIC GEOMETRY WEEK 12
MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f
More information3 Hausdorff and Connected Spaces
3 Hausdorff and Connected Spaces In this chapter we address the question of when two spaces are homeomorphic. This is done by examining two properties that are shared by any pair of homeomorphic spaces.
More informationMaths 212: Homework Solutions
Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then
More information1 Directional Derivatives and Differentiability
Wednesday, January 18, 2012 1 Directional Derivatives and Differentiability Let E R N, let f : E R and let x 0 E. Given a direction v R N, let L be the line through x 0 in the direction v, that is, L :=
More informationMASTERS EXAMINATION IN MATHEMATICS
MASTERS EXAMINATION IN MATHEMATICS PURE MATH OPTION, Spring 018 Full points can be obtained for correct answers to 8 questions. Each numbered question (which may have several parts) is worth 0 points.
More informationEconomics 204 Fall 2011 Problem Set 2 Suggested Solutions
Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit
More informationCOUNTABLE PRODUCTS ELENA GUREVICH
COUNTABLE PRODUCTS ELENA GUREVICH Abstract. In this paper, we extend our study to countably infinite products of topological spaces.. The Cantor Set Let us constract a very curios (but usefull) set known
More informationarxiv: v2 [math.ds] 21 Nov 2018
A Core Decomposition of Compact Sets in the Plane Benoît Loridant 1, Jun Luo 2, and Yi Yang 3 1 Montanuniversität Leoben, Franz Josefstrasse 18, Leoben 8700, Austria 2 School of Mathematics, Sun Yat-Sen
More informationSMSTC Geometry & Topology 1 Assignment 1 Matt Booth
SMSTC Geometry & Topology 1 Assignment 1 Matt Booth Question 1 i) Let be the space with one point. Suppose X is contractible. Then by definition we have maps f : X and g : X such that gf id X and fg id.
More informationMATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES
MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES PETE L. CLARK 4. Metric Spaces (no more lulz) Directions: This week, please solve any seven problems. Next week, please solve seven more. Starred parts of
More informationAustin Mohr Math 730 Homework. f(x) = y for some x λ Λ
Austin Mohr Math 730 Homework In the following problems, let Λ be an indexing set and let A and B λ for λ Λ be arbitrary sets. Problem 1B1 ( ) Show A B λ = (A B λ ). λ Λ λ Λ Proof. ( ) x A B λ λ Λ x A
More informationMath 3T03 - Topology
Math 3T03 - Topology Sang Woo Park April 5, 2018 Contents 1 Introduction to topology 2 1.1 What is topology?.......................... 2 1.2 Set theory............................... 3 2 Functions 4 3
More information