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Non-Obstructing Sets and Related Mappings* 1 NON-OBSTRUCTING SETS AND RELATED MAPPINGS* BY GORDON T. WHYBURN UNIVERSITY OF VIRGINIA Communicated August 6, 1960 1. Introduction.-In earlier papers' the type of set which separates no region in a locally connected space has been found to play a basic role, particularly when it constitutes the singular or exceptional set of points for a mapping. In what fol- lows, a further study of such sets will be made along with their connection with open and related mappings on sets having near-manifold structure. All spaces will be assumed separable and metric. Usually they are assumed to be locally connected generalized continua at least, i.e., connected, locally connected and locally compact as well as separable and metric. By a region in a space X is meant a connected open subset of X. 2. Almost uniform local connectedness and Q-sets.--A set X is said to be almost uniformly locally connected (abbreviated almost u.l.c.) provided that if C is any con- ditionally compact subset of X and e > 0, there exists 6 > 0 such that any two points x, y eC with p(x, y) < 6 lie together in a connected subset of X of diameter < E. Remark: It is readily verified that if X is the whole space, or is any closed set in the space, then X is almost u.l.c. if and only if it is locally connected. Now let X be a connected and locally connected space. A nondense closed set Q in X will be called a Q-set provided that if R is any region in X, then R - R * Q is connected, i.e., Q separates no region in X. (Note: "non-dense" means "contains no open set".) The following assertions result at once from the definition: (i) Q-sets are finitely additive (ii) 'Any closed subset of a Q-set is a Q-set (iii) A closed non-dense set Q is a Q-set if and only if each qe Q lies in an arbitrarily small region R in X such that R - R Q is connected. (iv) If X is an n-dimensional manifold, any ctosed subset of X of dimensionality < n - 2 is a Q-set. Relating the Q-sets with the property of almost uniform local connectedness is the (2.1) THEOREM. A non-dense closed subset Q of X is a Q-set if and only if X - Q is almost u. 1. c. For suppose Q is a Q-set and let a conditionally compact set C c X - Q and e > 0 be given. If there exists no 6 > 0 for the almost u.l.c. property, we readily find a sequence of point pairs (X,, yn) in C and a point q e Q such that both (X/) and (yj) converge to q but no connected subset of X - Q of diameter < e contains both Xn and yn. However, if R is a region in X of diameter < e containing q, R - R * Q is connected (since Q is a Q-set), lies in X - Q and contains both Xn and yJ for n sufficiently large. This contradiction proves X - Q almost u.l.c. On the other hand, suppose X - Q is almost u.l.c. but that for some region R in X, R - R * Q is the union of two separated nonempty sets R1 and R2. Then some q E Q is a limit point of both R, and R2, by connectedness of R. Let C consist of q + 1244 Downloaded by guest on September 25, 2021 VOL. 46, 1960 MA THEMATI7CS: G. T. WHYBURN 1245 ZX, + Zy,1, where (x,,) and (y,,) are sequences of points in RI1 and f2 respectively and each sequence converges to q. Let 3E be the distance from q to X - R and let a > 0 be given by almost u.l.c. of X - Q for C and e. Then p(x,,, y,,) < 5 and p(xnxq) < e > P(Yn, Q) for n sufficiently large. However, this gives a connected subset E of X - Q of diameter < e containing both Xn and y,,; and E c R and thus in R - R-Q, since p(xn, X - R) > 2e > 6(E). This is impossible because x. E Rf, yn E R2 and R1 and R2 are separated. Accordingly Q is a Q-set. (2.11) COROLLARY. The intersection of any two (or any finite number of) almost u.l.c. regions each dense in X is an almost u.l.c. region. (2.12) COROLLARY. If D is any uniformly locally connected region in a locally connected space X, the boundary of D is a Q-set in D. 3. q-mappings.-Let X and Y be connected and locally connected spaces. A closed mapping f(X) = Y will be called a q-mapping provided it is locally topolog- ical at all points of X - f-(F), where F is some Q-set in Y and f-I(F) is non-dense. It may be noted that, by known theorems, any light open and closed mapping on a 2 dimensional manifold (with or without boundary) is a q-mapping. It will fol- low from results below that a light q-mapping on a locally connected generalized continuum is necessarily open so that a light closed mapping on a 2-manifold is a q-mapping if and only if it is open. (3.1) THEOREM. If X and Y are localy connected generalized continua, the property of being a Q-set is invariant under any q-mapping f(X) = Y. For let Q be any Q-set in X and let F be a Q-set in Y such thatf is locally topologi- cal at all points of X -f-(F). Then if S is any region in Y, the set R = S -SSF is connected and thus is likewise a region in Y. Now f(Q) is non-dense. For if not, it would contain a region T not intersecting F and so that f-I (T) consists of a finite number of regions T1, T2, .. ., T, in X each mapping topologically onto T under F. This clearly is impossible, because Q. Ti is non-dense for each i whereas f(Q. T,) would have to contain T. Thus R - R f(Q) is non-empty. We show next that this set must be connected. If not, there exists a subset K of R f(Q) which is closed in R and so that R - K has at least two components R1 and R2 each having all of K on its boundary. Now if Q' = Q .f-I(K), Q' is locally compact, f(Q') = K and fj Q' is locally topological. Hence there exists a sequence of regions (Uj) in X such that f(RU)c ftUf is topological and the set Qj = Q'. Uj is compact, for each i, and Q' = EQj. Since K = 2f(Qi) and K is locally compact, for some n, f(Q,,) contains an open subset K' of K. Choose xeU,, so that f(x)eK' and let G be a region with xEG c G c U,, such that f(G) - (K - K') = P (the empty set). Then G -GC Q,, is connected, since Q,, is a Q-set. However, f(G -GC Qj) c R - K [since K' c f(Q,,) ] and inter- sects both R1 and R2, contrary to the fact that K separates R1 and R2 in R. (Note: If for zeG, f(z) e K, then f(z) e K' and only one point of U. can map onto f(z). Thus z must be in Qj). Thus R -Rff(Q) is connected. So also is the set R -Rff(Q) + (S - R) - (S - R) .f(Q), because S - R c F c P and both F and f(Q) are nondense. This set, however, is the same as S -S-f(Q). Since S was an arbitrary region in Y, we have shown that f(Q) separates no region in Y; and asf(Q) is closed and non-dense, it therefore is a Q-set. (3. 1 1) COROLLARY. q-mappings have the group property on locally connected Downloaded by guest on September 25, 2021 1246 MATHEMATICS: G. T. WHYBURN PROC. N. A. S. generalized continua, i.e., iffi(X) = Y and f2(Y) = Z are q-mappings, so also is f f2fl. For if Q is a Q-set in Y so that fi is locally topological at all points of X - f-I(Q) and fr-'(Q) is non-dense and E is a similar set in Z for f2, then the set F = f2(Q) + E is a Q-set in Z by (i) of §2 and the theorem; and since f-'(F) 3 fr-'(Q) and fif-'(F) v f2-'(E), f is locally topological on X-f-'(F). Alsof-1(F) is non-dense, because if not it would contain an open subset U of X - fg-(Q). However, f1(U) is then open and thus contains an open subset U' of Y - f2-'(E). Hence F would contain the open set f2(U'). A mapping f(X) = Y is quasi-open provided that if y e Y and U is any open set in X which contains a compact component of f` (y), then y is interior to f(U). (3.2) THEOREM. Let X and Y be locally connected generalized continua and let f(X) = Y be a mapping which is quasi-open on X - f-'(F) where F is some Q-set in Y.
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