Bounding Boxes for Compact Sets in $\calE^2$

Czeslaw Bylinski

Warsaw University, Bialystok

This work was partially supported by NSERC Grant OGP9207 and NATO CRG 951368.

Piotr Rudnicki

University of Alberta, Edmonton

This work was partially supported by NSERC Grant OGP9207 and NATO CRG 951368.

Summary.

We define pseudocompact topological spaces and prove that every compact space is pseudocompact. We also solve an exercise from [14]~p.225 that for a topological space $X$ the following are equivalent: \begin{itemize} \item Every continuous real map from $X$ is bounded (i.e. $X$ is pseudocompact). \item Every continuous real map from $X$ attains minimum. \item Every continuous real map from $X$ attains maximum. \end{itemize} Finally, for a compact set in $E^2$ we define its bounding rectangle and introduce a collection of notions associated with the box.

MML Identifier: PSCOMP_1

Contents (PDF format)

1. Preliminaries
2. Functions into Reals
3. Real maps
4. Pseudocompact spaces
5. Bounding boxes for compact sets in ${\calE}^2$

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