EPFL Logo
fr|en
Login
Concept

Pseudocompact space

In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the space be completely regular in the definition of pseudocompactness. Pseudocompact spaces were defined by Edwin Hewitt in 1948. For a Tychonoff space X to be pseudocompact requires that every locally finite collection of non-empty open sets of X be finite. 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 space is pseudocompact. The converse is true for metric spaces. As sequential compactness is an equivalent condition to compactness 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. Then X is neither compact, sequentially compact, countably compact, paracompact nor metacompact (although it is orthocompact). However, since X is hyperconnected, it is pseudocompact. This shows that pseudocompactness doesn't imply any of these other forms of compactness. For a Hausdorff space X to be compact requires that X be pseudocompact and realcompact (see Engelking 1968, p. 153).

Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications (3)
Related concepts (5)
Cocountable topology
The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable. It follows that the only closed subsets are X and the countable subsets of X. Symbolically, one writes the topology as Every set X with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of X. It is also T1, as all singletons are closed.
Metacompact space
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 open cover with the property that every point is contained only in finitely many sets of the refining cover. A space is countably metacompact if every countable open cover has a point-finite open refinement.
Glossary of topology
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology. All spaces in this glossary are assumed to be topological spaces unless stated otherwise. Absolutely closed See H-closed Accessible See . Accumulation point See limit point.
Show more

Copyright © 2025 EPFL, all rights reserved

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.