Pseudocompact space

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In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to \mathbb{R} is bounded.

[edit] Conditions for pseudocompactness

  • 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 the Heine-Borel theorem tells us that the compact subsets of \mathbb{R} are precisely the closed and bounded subsets.
  • If Y is the continuous image of pseudocompact X, then Y is pseudocompact. Note that for continuous functions g and h from X to Y and from Y to the real numbers, respectively, 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.
  • Notice that the definition of pseudocompactness is "unnesessarily weak": if X is pseudocompact and f is a continuous mapping from X to \mathbb{R}, then the image f(X) is in fact also closed, hence compact. This is true, since every non-closed set of reals is homeomorphic to unbounded subset of reals.

[edit] References

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