INVERSE PRESERVATION OF SMALL INDUCTIVE DIMENSION

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1 Voume 1, 1976 Pages INVERSE PRESERVATION OF SMALL INDUCTIVE DIMENSION by Peter J. Nyikos Topoogy Proceedings Web: Mai: Topoogy Proceedings Department of Mathematics & Statistics Auburn University, Aabama 36849, USA E-mai: ISSN: COPYRIGHT c by Topoogy Proceedings. A rights reserved.

2 TOPOLOGY PROCEEDINGS Voume INVERSE PRESERVATION OF SMALL INDUCTIVE DIMENSION Peter J. Nyiko8 The foowing resut has ong been known to Russians and is considered eementary, but the proof does not seem to have appeared in print: Theorem 1. Let X be a Hausdorff space and et f: X ~ Y be a perfect ight map. If Y is reguar, then ind X <ind Y. (A continuous function f is perfect if it is cosed and f-(y) is compact for a y E Y. It is ight if f-(y) is totay disconnected for a y E Y.) The proof makes use of the foowing trivia emma: Lemma 2. Let G and G be disjoint open subsets of a 2 space X and et K be a set whose cosure is contained in G U G " 2 Then Bd(K n G ) Bd K n G " In particu ar, if K is c open, so is K n G " Proof of Theorem 1. Let x be a point of X and et F = f-(f(x)). Let U be an open neighborhood of x. By zerodimensionaity of F, there exist disjoint cosed sets F and F 2 such that x E F C VI' F U F = F. Let VI and V be disjoint 2 2 open subsets of X containing F and F respectivey. Let 2 G = VI n V, G 2 V 2 Let V = G U G2 Because f is a cosed map, [f (Vc)]c is an open set containing f(x) whose inverse image is contained in V: The rest of the proof goes by induction. Suppose ind Y = o. Then there exists a copen set K containing f(x) and contained in [f(vc)]c. The inverse image of K is a copen set contained

3 64 Nyikos in Vi hence by the emma, f-(k) n G is copen, and we have x E f -1 (K) n G C G cu. Suppose the theorem has been proven for ind Y < n, and et ind Y = n+. By reguarity of Y, there exists a neighborhood A of f(x) whose cosure is contained in [f(vc)]c and whose boundary is of ind <no Since Bd f-(a) C f-(bd A) by continuity it foows that Bd f-(a) has sma inductive dimension <n by the induction hypothesis. By the emma, Bd f-(a) n G -1-1 Bd (f (A) n G ), so that f (A) n G is a neighborhood of x contained in G (hence in U) whose boundary has sma inductive dimension ~n, as was to be shown. The ony pace in the above proof where "perfect" was used was in getting disjoint cosed (and reative open) subsets of f-(y) into disjoint open subsets of X. This can be done in a number of aternative ways. For exampe (we take "reguar" and "norma" to incude "Hausdorff"): Theorem 2. Let X be a reguar space and et f: X ~ Y be a cosed map such that f-(y) is Lindeof (or ocay compact) and zero-dimensiona for a y E Y. If Y is reguar~ then ind X < ind Y. Theorem J. Let X be a norma space and et f: X ~ Y be a cosed map such that f-(y) is zero-dimensiona for a y E Y. Then ind X < ind Y. More generay, we have: Theorem 4. Let X be a topoogica space and et f: X ~ Y be a cosed map such that f-(y) is C*-embedded and zerodimensiona for a y E Y. If Y is reguar~ then ind X< ind Y. The foowing exampes show the necessity of "Hausdorff" in Theorem 1 and "norma" in Theorem 3.

4 TOPOLOGY PROCEEDINGS Voume S Exampe 5. Let X be the space consisting of a sequence of cosed and isoated points x n which converge to two distinct cosed points, x and z. Let Y be the space obtained by identifying x and z, and et f be the resuting map. (Ceary, Y is homeomorphic to w+.) Then f is a perfect ight map, and ind Y = 0, but ind X 1. Exampe 6. Let Z be a version of ~ [2, Exercise 5I] which is zero-dimensiona but not strongy zero-dimensiona [3] g:, Z + [0,1] be a continuous function such that g (0) and g (1) are not contained in disjoint copen sets. Let X be the space which is gotten by identifying g-() to a singe point and etting the neighborhoods of this point have a base consisting of the sets g-(-,]. Let the rest of X be given the reative topoogy as a subspace of Z. Then X is Tychonoff, and ind X = 1. Let f: X + Y be the map resuting from identifying a Let nonisoated points of X t~ a singe point, Y the resuting space (which is homeomorphic to w+). Then f is cosed, and f-(y) is cosed and zero-dimensiona for a y e Y. But ind Y = o. An interesting consequence of Theorem 1 is that the inverse preservation of a cass of zero-dimensiona spaces under perfect ight maps with Hausdorff domain, is equivaent to its inverse preservation under perfect maps with zero-dimensiona Hausdorff domain. Definition 7. Let a be a category of topoogica spaces and et ffi be a fu and repete subcategory of (1. Then fb is [Zighty] Zeft-fittinq in d if whenever f: X + [ight] map with X E C and Y E fb, then X E fb. Y is a perfect Theorem 8. Let ffi'be a category of aero-dimensiona Hausdopff spaces. The foowing are equivaent. (1) ffi is ighty eft-fitting in the categopy of Hausdopff

5 66 Nyi~os spaces. (2) ffi is eft-fitting in the category of zero-dimensiona Hausdorff spaces. (3) ffi is cosed hereditary, and every product of a space in ffi with a zero-dimensiona compact Hausdorff space is in ffi. Proof. That (1) is ~quiv~ent to (2) is immediate from Theorem 1. It is cear that (2) impies (3). To prove that (3) impies (2), one adapts the argument in [1], substituting "zerodimensiona" for "Tychonoff" and ~X for ax. Exampe 9. The category of N-compact spaces is ighty eft-fitting in the category of Hausdorff spaces. (A space is N-compact if it can be embedded as a cosed subspace in a product of countabe discrete spaces.) This foows from Theorem 8, since (3) is ceary satisfied. Probem 10. Let X be a Hausdorff space and et f: X ~ Y -1 be a perfect map such that ind f (y) <n for a y E Y. Is it true that ind X < ind Y+n? This is the natura generaization of Theorem 1, but the proof of Theorem 1 eans so heaviy upon the zero-dimensionaity of f-(y) that there seems itte hope of an affirmative answer here, even if we assume X and Y to be hereditariy norma. References [1] S. P. Frankin, On epi-refective hus, Gen. Top. App. 1 (1971), [2] L. Giman and M. Jerison, Rings of continuous functions, Princeton, Van Nostrand Co., [3] J. Teresawa, N U R need not be strongzy O-dimensionaZ, AMS Notices 23 (1976), A-296. Abstract 76T-G35. Auburn University, Auburn, Aabama 36830