Closed Maps and the Character of Spaces
نویسنده
چکیده
We give some necessary and sufficient conditions for the character of spaces to be preserved under closed maps. Introduction. Recall the following result due to K. Morita and S. Hanai [11] or A. H. Stone [14]: Let Y be the image of a metric space X under a closed map/. Then Y is first countable (or metrizable) if and only if every boundary df~x(y) of the point-inverse/"'^) is compact. E. Michael [8] showed that every 3/~x(y) is compact if Xis paracompact, and Fis locally compact or first countable. For the Lindelöfness of 3/"'(y) with X metric, see [15]. For a space X and x G X, let x(x, X) be the smallest cardinal number of the form | %(x) |, where ^(x) is a nbd base at x in X. The character x( X) of X is defined as the supremum of all numbers \(x, X) for x G X. Let/be a closed map from X onto Y. First, we show that the character of Y has an influence on the boundaries 3/~'(y); indeed, they become Lindelöf or a-compact by the situation of Y. Second, in terms of these boundaries, we give some necessary and sufficient conditions for the character of X to be preserved under/. We assume all spaces are regular and all maps are continuous and onto. 1. a-compactness of the boundaries. We recall some definitions. A space X is strongly collectionwise Hausdorff if, whenever D — {xa; a G A} is a discrete closed subset of X, there is a discrete collection {Ua; a G A) of open subsets with Ua D D = {xa}. Every paracompact space is strongly collectionwise Hausdorff. Let a > w0 and a+ be the least cardinal number greater than a. A space X is a-compact if every subset of X of cardinality a has an accumulation point in X. A space X is a-Lindelof if every open cover of X has a subcover of cardinality < a. Every a-Lindelöf space is a+ -compact. A space X is sequential if F C X is closed in X whenever Fil Cis closed in C for each compact metric subset C of X. If we replace "compact metric subset" by "countable subset", then such a space is said to have countable tightness. Every sequential space is precisely the quotient image of a metric space [3]. Received by the editors March 23, 1983 and, in revised form, July 11, 1983. 1980 Mathematics Subject Classification. Primary 54A25, 54C10; Secondary 54B15, 54D20, 54D55.
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