Topology Proceedings 28 No. 2 (2004), pp. 425-443

Maximal realcompact (and other) topologies

W.W. Comfort and A.W. Hager

Let RC be the class of realcompact Tychonoff spaces, m the first uncountable measurable cardinal (which is the first Ulam measurable cardinal), and P(m) the class of spaces in which each intersection of fewer than m open sets is open. We begin with a simple theorem: If X Î RC, there is another space mX on the same set as X whose topology is maximum for being RC and finer than the topology of X. (Of course, mX is discrete if and only if |X| < m.) This gives an operator m: RC ® M : = {X : mX=X} which is a coreflection. It is known that the P(m)-coreflection preserves RC; thus M Í P(m) ÇRC. The reverse inclusion represents an open question [Alan Dow has shown that this is undecidable. See the note at the end of this paper], but we prove it for two classes of spaces: Those for which the pseudocharacter does not exceed m, and those with fewer than m nonisolated points. Various examples of spaces in M are presented, indeed: for every cardinal number n there are spaces in M with exactly n nonisolated points. Actually, these observations about RC and m are but a special case. In the previous paragraph we may replace RC by any class R which is productive and closed-hereditary and contains the two-point space, while replacing m by s(R) : = sup{k: R contains the discrete space with k points} (which, it is known, is a measurable cardinal if not ¥).

Keywords: Maximal topology, measurable cardinal, epireflection, coreflection, realcompact space, P-space

Mathematics Subject Classification: 04A10 54A25 54B30 54D35 54D10 54G10 18A40 18B30

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Topology Proceedings, Volume 28, No. 2 (2004)
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