Ultrafilter Invariants in Topological Spaces

Abstract

Several topological properties are described using ultrafilter invariants, including compactness and perfect maps. If X is a Hausdorff space and D is a discrete space equipotent with a dense subset of X, then A-is a continuous perfect image of a subspace of ßD which contains D if and only if A" is regular. 1. Introduction. In [B,], A. Bernstein makes the following definitions. Let ^D be an ultrafilter over the positive integers N, let A' be a topological space, (xn:n G N) a sequence of points of X, and x G X. Then x is a ty-limit point of (xn: n G A/) provided that if t/ is open in X with x E U, then {«: *" G Í/} G 6D ; in this case we write x = ^-lim"^0O jc". If every sequence in X has a ty-limit point, then X is ^-compact. In [GS], John Ginsburg and the present author showed that a Hausdorff space X has all of its powers countably compact if, and only if, it iŝ D-compact, for some ^ in ßN \ N. In this paper, we generalize the notion of ^-limit to nets indexed by an arbitrary infinite cardinal. We show that any topological space is characterized by its generalized ^D-limits, and that a number of topological properties can be described by the use of ultrafilter invariants.

Ultrafilter Convergence in Ordered Topological Spaces

“…This is essentially a classical result (which holds for every topological space), due to [11] in the case ν = ω, and to [24] in the general case. See also [6] and [17] for other versions and generalizations.…”

“…Proof (a) ⇒ (b) If each X j satisfies any one of the conditions in Theorem 4.1, then, by the very same theorem, X j is D-compact. By an easy and classical property of D-compactness [4,24], every product of D-compact spaces is still D-compact, hence j ∈J X j is still Dcompact. Now notice that the proof that (1) implies any one of the conditions (1)-(2),(5)- (7) in Theorem 4.1 holds for an arbitrary topological space, not only for GO spaces.…”

“…By (2), every product of members of T , too, is [ν, ν]-compact, for every ν ∈ H . For every ν ∈ H , by [24] or [6,Theorem 3.4], there exists a ν-decomposable (equivalently, (ν, ν)-regular, since ν is regular) ultrafilter D ν such that every member of T is D ν -compact. It is easy to see that if κ is a regular cardinal, D is an ultrafilter, and the space κ is D-compact, then D is κ-descendingly complete (see the proof of [13,Proposition 1], though the result is stated with different terminology), equivalently, D is not κ-decomposable.…”

A characterization of the Menger property by means of ultrafilter convergence

“…For example, a regular topological space X is λ-bounded (i. e., the closure of every subset of cardinality ≤ λ is compact) if and only if X is D-compact, for every ultrafilter D over λ. In particular, a topological space is compact if and only if it is D-compact, for every ultrafilter D. See [Sa,Theorem 2.9, and Section 5].…”

The ℱ-Limit of a Sequence of Prime Ideals

“…Saks considered the sets X in a more general setting in his article [18], where he defined a closure operator by using -limit points for ultrafilters 's on arbitrary cardinal numbers.…”