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• W1993312126 abstract "We present simple examples of a realcompact zerodimensional space which is not Ar-compact and an N-compact space which is not strongly zerodimensional. Let N be the usual space of natural numbers. A topological space is called N-compact if it is homeomorphic to a closed subspace of some power of N. It is well known that realcompact strongly zerodimensional=> ./V-compact =^realcompact and zerodimensional. The converses are false, but known counterexamples are fairly complicated (cf. (1, 5-10)). The aim of this note is to present both counterexamples in a unified and simple way. It should be mentioned, however, that the spaces we construct are neither normal nor metacompact, and therefore they do not have any of the additional properties which the previously known examples have. Our construction has evolved from the ideas given in (2 and 7). As the referee has pointed out, the technique we use is close to that used by M. Wage in (11) in order to construct two strongly zerodimensional spaces whose product is not strongly zerodimensional. In fact, it turns out that our technique, although used for different purposes, is basically the same as that developed in (11). All undefined notions can be found in (3 and 4). We shall use in the sequel the ultrafilter characterization of A-compactness which states that a zerodimensional space X is iV-compact if and only if every clopen ultrafilter on X with the countable intersection property has nonempty intersection. The construction. Consider an arbitrary separable metric space X of cardi- nality c. Let D be an arbitrary countable dense subset of X. For every subset A of D with |clx A n clx(-D — A)| = c choose a point Xa in clx A n clx(L> — A) in such a way that xa / xa1 whenever A ■£ A'. Moreover, for every x from X — D let S(x) be a sequence in D converging to x such that if x = xa for some A G D then both A and D — A contain infinitely many points from S(x). Define a new topology on X by agreeing that all points from D are isolated and neighborhoods of x G X — D contain all but finitely many points from S(x). Denote the obtained space by X*." @default.
• W1993312126 created "2016-06-24" @default.
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• W1993312126 date "1984-04-01" @default.
• W1993312126 modified "2023-09-26" @default.
• W1993312126 title "Two easy examples of zero-dimensional spaces" @default.
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• W1993312126 doi "https://doi.org/10.1090/s0002-9939-1984-0760954-1" @default.
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