In mathematics, in the field of

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

, a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

is said to be pseudocompact if its image under any continuous function to R is

bounded Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...

. Many authors include the requirement that the space be

completely regular In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that i ...

in the definition of pseudocompactness. Pseudocompact spaces were defined by Edwin Hewitt in 1948.

Properties related to pseudocompactness

* For a

Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that i ...

''X'' to be pseudocompact requires that every

locally finite collection In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension. A collection of subsets of a topological spa ...

of non-empty

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...

s of ''X'' be

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past partici ...

. There are many equivalent conditions for pseudocompactness (sometimes some separation axiom should be assumed); a large number of them are quoted in Stephenson 2003. Some historical remarks about earlier results can be found in Engelking 1989, p. 211. *Every countably compact space is pseudocompact. For normal Hausdorff spaces the converse is true. *As a consequence of the above result, every

sequentially compact In mathematics, a topological space ''X'' is sequentially compact if every sequence of points in ''X'' has a convergent subsequence converging to a point in X. Every metric space is naturally a topological space, and for metric spaces, the noti ...

space is pseudocompact. The converse is true for

metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

. As sequential compactness is an equivalent condition to

compactness In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...

for metric spaces this implies that compactness is an equivalent condition to pseudocompactness for metric spaces also. *The weaker result that every compact space is pseudocompact is easily proved: the image of a compact space under any continuous function is compact, and every compact set in a metric space is bounded. *If ''Y'' is the continuous image of pseudocompact ''X'', then ''Y'' is pseudocompact. Note that for continuous functions ''g'' : ''X'' → ''Y'' and ''h'' : ''Y'' → R, the composition of ''g'' and ''h'', called ''f'', is a continuous function from ''X'' to the real numbers. Therefore, ''f'' is bounded, and ''Y'' is pseudocompact. *Let ''X'' be an infinite set given the

particular point topology In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The colle ...

. Then ''X'' is neither compact, sequentially compact, countably compact,

paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norm ...

nor metacompact (although it is orthocompact). However, since ''X'' is

hyperconnected In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...

, it is pseudocompact. This shows that pseudocompactness doesn't imply any of these other forms of compactness. * For a

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...

''X'' to be compact requires that ''X'' be pseudocompact and realcompact (see Engelking 1968, p. 153). * For a

''X'' to be compact requires that ''X'' be pseudocompact and metacompact (see Watson).

Pseudocompact topological groups

A relatively refined theory is available for pseudocompact

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two ...

s. In particular, W. W. Comfort and Kenneth A. Ross proved that a product of pseudocompact topological groups is still pseudocompact (this might fail for arbitrary topological spaces).Comfort, W. W. and Ross, K. A., Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16, 483-496, 1966

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Notes

References

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External links

* . * {{planetmathref, urlname=PseudocompactSpace, title=Pseudocompact space Properties of topological spaces Compactness (mathematics)

Properties related to pseudocompactness

Pseudocompact topological groups

Notes

See also

References

External links