The main purpose of this paper is to unify a number of theorems in topology whose conclusions state that a product of topological spaces has a compactness-like property.Three such theorems are (1) the Tychonoff theorem: Every product of compact spaces is compact, (2) the theorem of C.T. Scarborough and A.H. Stone: Every product of at most N1 sequentially compact spaces is countably compact, and (3) the theorem of N. Noble: A countable product of Lindelöf P-spaces is Lindelöf.
Theorem. Let X be a T1 space. The following are equivalent:
X has a σ-disjoint base.
X is quasi-developable and has a base that is the union of a sequence of rank 1 collections.
X has a quasi-development (n) with the property that for each x, , n a positive integer} is a base for (x).
Theorem. Let X be a T1 space. The following are equivalent:
X has a σ-point finite base.
X has a quasi-development (n) with each n well ranked.
X has a quasi-development (n) with each n Noetherian of sub-infinite rank.
X has a quasi-development (n) with each n Noetherian of point finite rank.
It is known that if X is a compactum and Y is metrizable Sh5(X × Y) is not generally determined by Sh5(X) and Sh5(Y), where Sh5(Z) is the strong shape of Z in the sense of Borsuk. In this paper it is proved that Sh(X × Y) is uniquely determined by Sh(X) and Sh(Y), where Sh(Z) is the shape of Z in the sense of Fox. If X is an FANR and Y is an MANR, then X × Y is an MANR.
A previous paper constructed exponential laws in the category TopB of spaces over B. The present paper relates these laws to constructions known for locally trivial maps, and constructs also new exponential laws for ex-spaces, fibred section spaces and fibred relative lifting spaces. Versions of these laws for homotopy classes of maps are discussed.
We provide a general framework for the study of the finest linear (locally convex) topology which coincides on a family of subsets with a given linear (locally convex) topology. It is proved that the formation of such topologies always commutes with linear direct sums. We characterize the corresponding situation for products and prove a result about locally convex direct sums sufficiently general to cover the examples which already occurred in the literature. Moreover the 0-nbhd. filters of such topologies are characterized, and several examples are considered.
De Groot and Verbeek have both asked for an example of a compact Hausdorff space which is not supercompact. Is is shown here that if X is not pseudocompact, then βX is not supercompact. It is done in the more general setting of Wallman compactifications.
Characterization of countably compact, Lindelof, H-closed, first countable and second countable are provided in terms of the nearness structure. Applications of these results are provided for uniform spaces and specific nearness structures.
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