Journal of Formalized Mathematics
Volume 9, 1997
University of Bialystok
Copyright (c) 1997 Association of Mizar Users

## 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

The terminology and notation used in this paper have been introduced in the following articles [20] [23] [1] [22] [16] [7] [18] [10] [21] [24] [3] [4] [13] [12] [15] [11] [19] [17] [6] [5] [2] [8] [9]

#### 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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