function provided the associated graph function g:x -) X X Y defined

Transcription

1 QUASI-CLOSED SETS AND FIXED POINTS BY GORDON T. WHYBURN UNIVERSITY OF VIRGINIA Communicated December 29, Introduction.-In this paper we develop new separation and intersection properties of certain partially closed sets and apply these to obtain an elementary simple proof for the Hamilton-Stallings extension of the Brouwer fixed-point theorem to peripherally continuous and connectivity functions (see refs. 1 and 2). A set E in a topological space X is quasi-closed, or of external dimension 0, provided that for each p e X - E the set E + p has Menger-Urysohn dimension 0 at p, i.e., any neighborhood of p contains an open set about p whose boundary does not meet E (see ref. 3). A set G in X is quasi-open provided its complement is quasiclosed, or equivalently, provided that for any p E G and any open set U containing p there is an open set about p in U whose boundary lies in G. A function f:x -- Y is peripherally continuous provided that for any x E X and any open sets U in X about x and V in Y about f(x), there exists an open set W with x e W c U and whose boundary Fr(W) maps into V under f. Also f is a connectivity function provided the associated graph function g:x -) X X Y defined by g(x) = [x,f(x)] preserves connectedness of sets, i.e., the graph r[f C) of the restriction of f to any connected set C c X is connected. It is known and easy to prove (see ref. 3) that a function f is peripherally continuous if and only if the inverse of every closed (open) set in Y is quasi-closed (quasi-open). Also it is known that for a very large class of domain and range spaces X and Y, including all Euclidean manifolds of dimension > 2, peripherally continuous functions coincide with connectivity functions (see refs. 4 and 5; also see results below). From now on, we assume that X and Y are regular T,-spaces. A space X is locally cohesive (see ref. 3) provided it is connected and each open set about a point x of X contains the closure of a canonical region about x, i.e., a connected open set R having a connected boundary Fr(R) and such that R is unicoherent between x and Fr(R) (equivalently, X is unicoherent between x and X- R). Obviously any locally cohesive space is locally connected and has no local cut point. Further, it is an immediate consequence of the unicoherence assumption that if W is a canonical region in X about a E X, then any set K separating a and Fr(W) in It contains the boundary of a canonical region R about a lying in W. Thus, in particular, in a locally cohesive space X, if E is any quasi-closed set in X, any open set U in X abou. a e X - E contains a canonical region R about a with R c U and E-Fr(R) = 44 Equivalently, if G is quasi-open, any open set U about a E G contains a canonical region R about a with R c U and Fr(R) c G. 2. Separation Theorems.-Two subsets A and B of a connected space X are said to be weakly separated in X by a set E provided no component of X - E meets both A and B. The sets A and B are separated in X by E provided X - E is the union of two disjoint open sets, one or both of which may be empty, containing A and B, respectively. Note that if both A and B are nonempty, in either case E must also be nonempty. -THEOREM (2.1). Let A and B be disjoint nondegenerate closed and connected sets 201

2 202 MATHEMATICS: G. T. WHYBURN -PR-oc. N. A. S. in a locally cohesive space X. Any quasi-closed set L which weakly separates A and B in X contains a closed set K (nonempty) which separates A - K and B - K in X. Proof: If either A or B is contained in L, we have only to take K = A (resp., B). Thus we suppose neither A nor B lies in L. Let H be the union of all components of X - L intersecting A and let V be the union of all components of X - (12 + A) intersecting B. Then let K = Fr(V) + FI-B and define U = X - (V + K). Clearly U and V are disjoint and open, K is closed, and K c H + A -L. These definitions and relations give X-K= U+V and B-KcB-FtcCV. (i) Also, since A. V C K, we have A-KcX-(V+K) = U. (ii) Thus we have left only to prove K = Fr(V) + 1.B C L. (iii) Suppose, on the contrary, that some x e K is not in L. Let R be a canonical region about x so the R contains neither A nor B, intersects A or B only in case x is in A or B, respectively, and so that the boundary C of R lies in X - L. Then in any case C intersects and thus lies in some component Q of X - L in H. For x in A, this is true because C meets A; and for x not in A, it follows because some component of X - L in H intersects both R and A. However, for x in B, this is impossible since C also meets B; and for x not in B, it is also impossible because C then must meet some component of V. This proves (iii). THEOREM (2. 2). If X is loally cohesive, any connected set in X lying in the union of two disjoint quasi-open sets lies entirely in one of them. Proof: Suppose E is connected and lies in U + V, where U and V are disjoint quasi-open sets, and suppose there exist points a e E- U and b e E. V. For each x e E let Q. be a canonical region about x whose boundary C, lies in U or in V accordingly as x is in U or in V. Since E is connected, there is a simple chain of these regions ae Q1,Q2..,Qn D b from a to b only the first of which contains a and only the last contains b. Let Ci = Fr(Qj) for each i. Then for 1 < k < n, both Qkand Qk+l must intersect Ck because each of them meets Qk but is not contained in Qk. Thus Ck must intersect both Ck-l and Ck+1 because Ck is connected and Qk-1 Qk+1 = (. However, it follows from this that Ci c U for all i = 1,... ^ contrary to Cn C V. For C, c U and C2 intersects C1, giving C2 C U, and so on to C". COROLLARY (2.21). Any peripherally continuous function f:x Y of a locally cohesive space X into a completely normal T1 space Y preserves connectedness; and thus it is a connectivity function whenever X X Y is completely normal (see refs. 4 and 5). For let E be a connected set in X. If f(e) were not connected, there would exist disjoint open sets U' and V' in Y each intersecting f(e) and withf(e) c U' + V'. However, U = f-1(u') and V = f-1(v') would then be disjoint quasi-open sets each meeting E. Using (2.1) we now obtain a basic extension of the Hurewicz-Wallman intersection result for closed sets in the unit interval In of Euclidean n-space E' (see p. 40

3 VOL. 57, 1967 MATHEMATICS: G. T. WHYBURN 203 of ref. 6). For each i, 1 < i < n, let Ai and B1 be the faces of In on which xf = 0 and xi = 1, respectively. THEOREM (2.3). Given quasi-closed sets C1,C2...,C. in In such that for each i, 1 < i < n,ciweakly separatesai andbiinl". ThennflC $. 1 Proof: (Compare pp. 40 and 41 of ref. 6.) For each i, by (2.1), Ct contains a closed set K1 which separates A, - K1 and B1 - K1 in In, so that In - K1 = U1 + Vi, where U1 and VL are disjoint and open and contain A1 - K1 and Bi - Ki, respectively. Now define a function f(x), x e In, by letting f(x) be the terminal end of the position vector x + d(x) in En, where the ith component di of the vector d(x) is 4t p(x,ki), the sign being + for x e U1 and - for x e V1. Then for x e Ui and each i, we have di = p(x,ki) < 1 - xso that 0 < xi + di < 1; and for x e Vi, di = -p(x,ki) > - xand again 0 < xi + d.<xt < 1. Thus in any case f(x) e In. Since f clearly is continuous and f:in -O In, we have f(xo) = xo for some Xo e In by the Brouwer fixed-point theorem. Thus d(xo) = 0 and xo e f Ki c n C, FIXED-POINT THEOREM (Hamilton-Stallings). Any peripherally continuous function of In into itself, n > 2, has at least one fixed point. 7 he same is true of any connectivity function of In into itselffor n > 1. Proof: Let f.:in In, n > 2, be any peripherally continuous function. We visualize In X In in the form In X I"n where In = Ih X 12 X... X In and I", = I'l X I'2 X... X I'n; and for x = (Xl, X2,..., X7n) E In let f(t) = x' = (X'IX'2,. x'7n) e I".. Let g In X be the graph function for f, i.e., g(x) = [x~f(x)]; and for 1 < i < n, let p1 = 7r4g.In7. I, X 1', be the projection of the graph of f into the plane I, X 1'm given by pi(x) = 7r1 (x,x') = (xi,x'i) e Ii X I,, where f(x) = x = (x'x'2,n..,x'n). Let Ai be the diagonal xi = x'i in the plane Ii X I'i. That f has a fixed point now follows from three assertions: (i) For each i, 1 < i < n, pi is peripherally continuous. (ii) For each i, 1 < i < n, the set C1 = pi-'(ai) is quasi-closed aiid it weakly separates in In the faces A1 and Bi of In on which xi = 0 and xi = 1, respectively. n (iii) n Ci # b andf(x) = x for eachxe fl Cn,. 1 1 To verify (i), let E be any closed set in Il X I'i. Then 7r-'(E) is closed since 7r1 is continuous. Thus the set g-l 7r-'(E) = pj-(e) is quasi-closed, sincef is peripherally continuous and thus so also is its graph function g. Whence, pi is peripherally continuous. To prove (ii) note that since Ai is closed, C1 is quasi-closed by (i). Now if some component Q of In _ C2 intersected both Ai and Bi, pi (Q) would be connected by (2.21). However, p,(q) would then meet Ai because it contains a point pi(a), a e Ai, where x'i > xi and also a point pi(b), b e B1, where x'1 < xi. Assertion (iii) now follows from (ii) and (2.3), and this concludes the proof of the first statement in our theorem. The second statement is verified by the standard simple argument in case n = 1. For n> 2 it results from the first statement together with the following result of Hamilton and Stallings. THEOREM. If X is locally compact, locally cohesive and metric, any connectivity function f:x -- Y of X into a regular Ti-space Y is peripherally continuous.

4 204 MATHEMATICS: G. T. WHYBURN Patoc. N. A. S. For completeness we give a proof, which is a simplified form of the one given earlier by the author (ref. 3). Proof: For x e X let U and V be open sets about x and f(x), respectively. We may suppose U chosein as a canonical region with a compact closure. Thus the boundary B of U is connected and U is unicoherent between x and B. Let U1 and V1 be open sets about x and f(x), respectively, with L1 c U, VP c V. Let D = U1 f-1( fo. Then D is semiclosed (see refs. 2 and 7). Thus the decomposition of U into the components of D, the set B and single points of U - D is upper semicontinuous. Hence its natural mapping 0: U -M (onto) is monotone and closed and thus AIl is a locally connected continuum which is unicoherent between a = +(x) and b = +(B). Now a and b cannot lie in a true cyclic element E of Ml; for if so, E - O(D) = Q' is connected because +(D) is totally disconnected and E is unicoherent. However, Q' + a is then connected, and thus so also is Q + A, where Q = 0-'(Q%), x e A = 0-1(a). This is impossible because A X f(x) is compact and also is open in r Ff Q + A), since (Ul X V1)Ir' f Q + A) = A X f(x). Thus some point z E A1 separates a and b in A1 and p-1(z) = Z is a component of D, since U can have no cut point. Thus if we let R' be the component of M -z containing x, then 0-1(R') = R is an open set about x with R C U and f [Fr(R)] = f(z) c f(d) c I1 c V. 4. Concluding Rfemarks.-(1) The proof given in 3 for the fixed-point theorenm has considerable in common with both Hamiltonas and Stallings' versions of the argument. However, it follows more closely the original ideas and spirit of Hamilton's form. It avoids all approximation and subdivision techniques and makes no use of almost continuity as introduced and used by Stallings. (2) Theorem (2.1) in 2 remains valid when the requirement that A and B be nondegenerate is deleted. Suppose A reduces to a single point a. If a e L, we have only to take K = a, U = 4X, and V = X - a. If a is not in L, we can replace a by the closure Wf of some region W inl X about a with WTn B = 0. For let R be a canonical region about a with R c X - B and whose boundary C does not meet L. Then if for every region W about x with It C R there existed a component Q of X - L intersecting both W and B, we would have Q D C for every such W and thus Q is the same for all such W. However, a would then be a limit point of Q and Q + a would be a connected subset of X - L meeting both A and B. Thus L weakly separates WT and B for some such W. Similarly, if B reduces to a point b not, in L, we can replace b by the closure S of a region S with S W = 4 so that L weakly separates WT and S in X. (3) Likewise it is clear that the condition that A and B be connected could be replaced by the requirement that each of them be of dimension > 0 at each point. 1 Hamilton, 0. H., "Fixed points for certain non-continuous transformations," Proc. Am. Math. Soc., 8, (1957). 2 Stallings, J., "Fixed point theorems for connectivity maps," Fund. Math., 47, (1959). 3 Whyburn, CG. T., "Loosely closed sets and partially continuous functions," to appear in Mich. Math. J. 4 Hagen, M. P., "Equivalence of connectivity maps and peripherally continuous transformations," Proc. Am. Math. Soc., 17, (1966). 6 Whyburn, G. T., "Connectivity of peripherally continuous functions," these PROCEEDINGS, 55, (1966).

5 VOL. 57, 1967 MATHEMATICS: G. T. WHYBURN Hurewicz, W., and H. Waltman, Dimension Theory (Princeton: Princeton University Press, 1941). 7Hagan, M. R., "Upper semicontinuous decompositions and factorization continuous transformations," Duke Math. J., 32, (1965). of certain nlonl-