Minimal First Countable Topologies

Characterizations of functionally compact spaces

“…A space (X, T) is called ./irsf countable and minimal Hausdorff if r is first countable and Hausdorff and if n o first countable topology o n X which is strictly weaker than T is Hausdorff (Stephenson (1969)). In our next theorem we give a characterization of first countable minimality.…”

Minimal first countable Hausdorff spaces

“…The following characterization theorems may, therefore, be of interest. In order to obtain a if-analogue of Theorem 2.4 of [11], we need a second definition.…”

“…For the case & -£$f, the following two results, established in [11], will be used below. An <^-space X is ^g^-closed if and only if every countable open filter base on X has nonempty adherence.…”

“…Suppose that J^ is a countable open filter base on X{^) and no point of X is an adherent point of ^7 A slight modification of the proof of Lemma 2.17 in [11] shows that there exists a free, countable, pseudocompact, zero-dimensional filter base gΌnl which is stronger than the filter base &~\X. By the maximality of there exists ^T* e ^ with G Π Hnonempty for all G e gf and He Thus p{3ίΓ) is an adherent point of To check semiregularity, it suffices to observe that if …”

Maximal Extensions of First-Countable Spaces

“…Let us call a first-countable space maximal, or, more precisely, maximal with respect to first-countability, if it has no proper, first-countable extension. According to [S,Theorem 2.9], a first-countable space is maximal if and only if it is pseudocompact. (Note that our maximal spaces are identical to Stephenson's first countableand completely regular-closed spaces.)…”