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

Generic use

The expression ''P-space'' might be used generically to denote a topological space satisfying some given and previously introduced topological invariant ''P''. This might apply also to

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of a different kind, i.e. non-topological spaces with additional structure.

''P-spaces'' in the sense of Gillman–Henriksen

A ''P-space'' in the sense of Gillman– Henriksen is a topological space in which every countable

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

of open sets is open. An equivalent condition is that countable unions of

closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...

s are closed. In other words, G_δ sets are open and F_σ 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 In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...

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. 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 space In the mathematical field of topology, a locally finite space is a topological space in which every point has a finite neighborhood, that is, an open neighborhood consisting of finitely many elements. A locally finite space is Alexandrov. A T1 s ...

s, which include finite spaces and

discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...

''P-spaces'' in the sense of Morita

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''.

''p-spaces''

A notion of a ''p-space'' has been introduced by Alexander Arhangelskii.Encyclopedia of General Topology, p. 278.

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