In the mathematical field of topology, there are various notions of a P-space and of a p-space.

Generic useEdit

The expression P-space might be used generically to denote a topological space satisfying some given and previously introduced topological invariant P.[1] This might apply also to spaces of a different kind, i.e. non-topological spaces with additional structure.

P-spaces in the sense of Gillman–HenriksenEdit

A P-space in the sense of GillmanHenriksen is a topological space in which every countable intersection of open sets is open. An equivalent condition is that countable unions of closed sets are closed. In other words, Gδ sets are open and Fσ sets are closed. The letter P stands for both pseudo-discrete and prime. Gillman and Henriksen also define a P-point as a point at which any prime ideal of the ring of real-valued continuous functions is maximal, and a P-space is a space in which every point is a P-point.[2]

Different authors restrict their attention to topological spaces that satisfy various separation axioms. With the right axioms, one may characterize P-spaces in terms of their rings of continuous real-valued functions.

Special kinds of P-spaces include Alexandrov-discrete spaces, in which arbitrary intersections of open sets are open. These in turn include locally finite spaces, which include finite spaces and discrete spaces.

P-spaces in the sense of MoritaEdit

A different notion of a P-space has been introduced by Kiiti Morita in 1964, in connection with his (now solved) conjectures (see the relevant entry for more information). Spaces satisfying the covering property introduced by Morita are sometimes also called Morita P-spaces or normal P-spaces.


A notion of a p-space has been introduced by Alexander Arhangelskii.[3]


  1. ^ Aisling E. McCluskey, Comparison of Topologies (Minimal and Maximal Topologies), Chapter a7 in Encyclopedia of General Topology, Edited by Klaas Pieter Hart, Jun-iti Nagata and Jerry E. Vaughan, 2003 Elsevier B.V.
  2. ^ Gillman, L.; Henriksen, M. (1954). "Concerning rings of continuous functions". Transactions of the American Mathematical Society. 77 (2): 340–352. doi:10.2307/1990875. JSTOR 1990875. Cited in Hart, K.P. (2001). "P-point". In Hazewinkel, Michiel (ed.). Encyclopaedia of Mathematics, Supplement III. Kluwer Academic Publishers. p. 297. ISBN 1-4020-0198-3.
  3. ^ Encyclopedia of General Topology, p. 278.

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