Normality-type Properties in exp(X)
by
A.P. Kombarov
Moscow State University

All spaces are assumed to be T₁. Recall that a space is said to be pseudonormal if every countable closed subset has arbitrarily small closed neighborhoods. A space is called to have property D [4] if every countable closed discrete set has arbitrarily small closed neighborhoods. A regular space is said to have property wD [4] if every infinite closed discrete set has an infinite subset which has arbitrarily small closed neighborhoods. Clearly, normal ===> pseudonormal ===> D ===> wD. Let Q be a topological property. A space X is said to be point-Q (Ñ-Q), if for every x in X a subspace X \{x} the subset X² \\Delta, where \Delta = {(x, x) : x in X} is a diagonal) has the property Q. The exponential space exp(X) is the set of all non-empty closed subsets of X with Vietoris (finite) topology.

Theorem 1 If exp(X ) is a point-D space, then X is a hereditarily separable perfectly normal countably compact space.

Corollary 1 If exp(exp(X)) or exp(X ×X) is point-D, then X is a metrizable compact space.

Corollary 2 If exp(X) is hereditarily pseudonormal, then X is a hereditarily separable perfectly normal countably compact space.

We note here that a space X is a metrizable compact space if exp(X) is hereditarily normal [3] or is regular hereditarily countably paracompact [5]. Every regular countably paracompact space is pseudonormal. Thus the next problem seems to be natural.

Problem 1 Is X a metrizable compact space, if exp(X) is hereditarily pseudonormal?

Theorem 2 Let X be a compact space. Then exp(X) is point-wD if and only if X is hereditarily separable and perfectly normal.

Theorem 3 If exp(X) is Ñ-normal, then X is a hereditarily separable perfectly normal compact space.

Problem 2 Is X a metrizable compact space, if exp(X) is Ñ-normal?

If exp(X) is regular and perfect (=every closed set is a G_\delta), then X is a metrizable compact space [7]. The next theorem 4 is an extension of this theorem.

Theorem 4 If exp(X) is Hausdorff and \omega-perfect (=every separable closed set is a G_\delta) , then X is a metrizable compact space.

A space X is said to be weakly normal [1, 2], if for every two disjoint closed subsets A and B of X there exists a continuous mapping f of X into R^\omega such that images of A and B are disjoint. Clearly, every normal space is weakly normal. If there exists a one-to-one continuous mapping of X onto a separable metrizable space, then the space X is weakly normal. So the Niemytzkij plane or the square of the Sorgenfrey line can serve as examples of weakly normal spaces which are not normal [1]. Can one use weak normality instead of normality in Velicko's theorem [8] (and in Coban's theorem[3]): if exp(X) is ( hereditarily ) normal , then X is a( metrizable) compact space? It is easy to see that this is not the case. Indeed there exists a one-to-one continuous mapping of exp(\omega) onto D^\omega, so exp(\omega) is hereditary weakly normal, but the space \omega is not compact.

Theorem 5 If X is a countably compact space and if exp(X) is weakly normal, then X is compact.

Theorem 6 If X is a countably compact space and if exp(X) is hereditarily weakly normal, then X is a perfectly normal hereditarily separable compact space.

Corollary 4 If X is countably compact and exp(exp(X)) or exp(X ×X) is hereditarily weakly normal, then X is a metrizable compact space.

Problem 3 Is a compact space X metrizable, if exp(X) is hereditarily weakly normal ?

And the final theorem 7 is a slight generalization of Noble's theorem from [6].

Theorem 7 All powers of a T₁-space X are weakly normal if and only if X is compact T₂.

1 A.V.Arhangel'skii Divisibility and cleavability of spaces Recent Developments of General Topology and its Applications, Berlin, Math. Research 67 1992 13-26
2 A.V.Arhangel'skii A survey of cleavability Topology Appl. 54 1993 141-163
3 M.M.Coban Note sur topologie exponentielle Fund. Math. 171 1971 27-41
4 E.K. van Douwen The Integers and Topology in: K.Kunen and J.E.Vaughan, eds., Handbook of Set-Theoretic Topology (North-Holland, Amsterdam, 1984) 111-167
5 A.P.Kombarov On F_\sigma-countably paracompact spaces Moscow Univ. Math. Bull. 44 1989 98-101
6 N.Noble Products with closed projections II Trans. Amer. Math. Soc. 160 1971 169-183
7 V.V.Popov On a space of closed subsets DAN SSSR 229 1976 1051-1054 in Russian
8 N.V.Velicko On a space of closed subsets, Siberian Math. Journ. 16 1975 484-486

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-55.