The concept of p-space is quite recent. It was introduced by Arhangel'skii [2]. The definition of p-space given in [2] involves compactification of the space. In view of the interesting properties of p-spaces obtained in [2], Alexadroff [1] suggested a problem of finding a direct intrinsic definition (without appeal to compactification). The main aim of this note is to answer the above problem.I am grateful to Dr. S. K. Kaul for his comments.

Let f (n) be the minimum number of arcs among oriented graphs of order n and diameter 2. Here it is shown for n

This note is closely related, as far as methods are concerned, to [3]. In [3] Ponomarev established “In order for a regular space X to be Lindelöf, it is necessary and sufficient that for each open covering ω of the space X there exists an ω-mapping ƒ:X → Y onto some separable metric space Y.” It is the purpose of this note to show that if the word “countable (or finite)” is inserted in the proper place we can obtain an analogous characterization for normal countably paracompact (or normal) spaces.

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In [1] Arhangel'skiĭ announced that any σ-paracompact p-space could be mapped onto a Moore space by a perfect map. However Burke [3] recently showed that this is not true in general and he gave an example of a T2, locally compact, σ-paracompact space which cannot be mapped onto a Moore space by a perfect map.