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If 𝒫 is a property of topologies,
a 𝒫-space (X,𝒯−) is called a 𝒫-minimal space if there exists no 𝒫-topology on X
properly contained in 𝒯 . Throughout the following, ℋ =first countable and Hausdorff
and 𝒞 = first countable and completely Hausdorff (a space X is called completely
Hausdorff if the continuous real valued functions defined on X separate the points of
X).
In this paper we give examples of ℋ-minimal 𝒞-spaces that are (i) not regular and
(ii) regular but neither completely regular nor countably compact.
Two other results obtained are the following. (a) Every locally pseudocompact
zero-dimensional ℋ-space can be embedded densely in a pseudocompact
zero-dimensional ℋ-space. (b) Let 𝒫 = 𝒞, completely regular ℋ, or zerodimensional ℋ,
and suppose that X is a 𝒫-space such that for every 𝒫-space Y and continuous mapping
f : X → Y,f is closed. Then X is countably compact.
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