Some applications of ultrafilters in topology

Category in function spaces. I

“…In this section, let Σ be the class of all collection wise Hausdorff [3] spaces X in which the set X d -{x e X: x is not isolated} is a discrete subspace of X. Other characterizations of members are given in our next lemma, whose proof is elementary.…”

“…Furthermore, the pseudonormality hypothesis in 8.4 can be replaced by any hypothesis which enables (continuous) functions on countable closed discrete subspaces to be extended to continuous functions on all of X. One such hypothesis is that X be strongly collectionwise Hausdorff [3] and completely regular. Examples show that some hypothesis beyond complete regularity is needed to prove 8.4 since there is a completely regular pseudocompact space X in which each countable set is closed [4, 5.1 and 5.3] and for such an X, C π iX) cannot even be a Baire space (cf.…”

A Countably Compact Space and Its Products

“…To complete the proof, let / be the extension of / to ß (co). Since r = r-lim n, and /•-limits are preserved by continuous maps [5], we have f(r) = r-lim f(n) = r-lim/(«) = x. Thus t(x) < t(r).…”

“…For other uses of types of ultrafilters in topology, see the interesting papers by J. Ginsburg and V. Saks [5] and V. Saks [9].…”

“…For an ultrafilter r in ß (of) \ to and a sequence {xn: n < to} in a topological space X, we say that x£lis the r-limit of {xn: n < to} (denoted x = r-lim xn) provided that for every neighborhood U of x we have {n: x" E U) E r. A space X is called r-compact if every sequence in X has an r-limit. Bernstein proved that if a space X is r-compact, then XK is r-compact (hence countably compact) for every cardinal k [1, Theorem 4.2] (the converse was proved by Ginsburg and Saks [5,Theorem 2.6]). To prove that our space X satisfies condition (1.1) we show that X is r-compact for certain r (note that X is not r-compact for all r because X is not to-bounded [1, Theorem 3.5]).…”