The structure of the poset of regular topologies on a set

Abstract

We study the subposet Σ3(X) of the lattice L1(X) of all T1-topologies on a set X, being the collections of all T3 topologies on X, with a view to deciding which elements of this partially ordered set have and which do not have immediate predecessors. We show that each regular topology which is not R-closed does have such a predecessor and as a corollary we obtain a result of Costantini that each non-compact Tychonoff space has an immediate predecessor in Σ3. We also consider the problem of when an R-closed topology is maximal R-closed.2010 MSC: Primary 54A10; Secondary 06A06, 54D10

On cellular-compactness and related properties

“…A first countable regular feebly compact topology is maximal among regular feebly compact topologies (see [2]) and it follows immediately that a first countable regular cellularcountably-compact topology is maximal among regular cellular-countably-compact topologies. However, the -product of Example 4.3 is a Fréchet cellular-countably-compact Tychonoff space which is not maximal (nor even T -maximal) cellular-countably-compact (nor maximal cellular-sequentially-compact) since for any p ∈ , \{ p} is cellular-compact and hence \{ p} ⊕ {p} is a stronger cellular-sequentially-compact Tychonoff topology on .…”

“…(2)⇒(1) Suppose that X has the property described in (2) and let U be a family of disjoint non-empty open subsets of X . Putting F = U ∪ {X \cl( U)}, we have that F is dense in X .…”

Reflecting properties in continuous images of small weight

“…It follows immediately from [13,Proposition 3.3 (b)] that if X has property P, then it is perfect and so ψ(X ) = ω. Since both X and Y are pseudocompact, a folklore result which appears in a slightly stronger form as [1,Lemma 3.7] then implies that χ(X ) = χ(Y ) = ω also.…”