Vol. 36, No. 3, 1971

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Minimal first countable Hausdorff spaces

Robert Moffatt Stephenson Jr.

Vol. 36 (1971), No. 3, 819–825

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, =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 -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 -space can be embedded densely in a pseudocompact zero-dimensional -space. (b) Let 𝒫 = 𝒞, completely regular , or zerodimensional , 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.

Mathematical Subject Classification
Primary: 54.20
Received: 17 February 1970
Published: 1 March 1971
Robert Moffatt Stephenson Jr.