mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 ...

, 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 poin ...

is said to be pseudocompact if its image under any

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...

to R is bounded. 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 is ...

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 is ...

''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 space X ...

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 are su ...

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, a verb form that has a subject, usually being inflected or marke ...

. There are many equivalent conditions for pseudocompactness (sometimes some

separation axiom In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometim ...

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 space In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...

s 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 notio ...

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 settin ...

. 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 Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...

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 collect ...

. 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 normal ...

nor

metacompact In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an op ...

(although it is

orthocompact In mathematics, in the field of general topology, a topological space is said to be orthocompact if every open cover has an interior-preserving open refinement. That is, given an open cover of the topological space, there is a refinement that is ...

). 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 ma ...

''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 st ...

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