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Theorem 6.20 (Heine-Borel Theorem). Let A be a subset of R" with the standard topol- ogy. Then A is compact if and only if A is closed and bounded.

Theorem 6.20 (Heine-Borel Theorem). Let A be a subset of R" with the standard topol- ogy. Then A is compact if and only if A is closed and bounded.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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Could you explain how to show 6.20 in detail?

**Theorem 6.20 (Heine-Borel Theorem).** Let \( A \) be a subset of \( \mathbb{R}^n \) with the standard topology. Then \( A \) is compact if and only if \( A \) is closed and bounded. This theorem is a key result in real analysis and topology, connecting the concepts of compactness, closed sets, and boundedness in Euclidean spaces. It establishes that a subset of a Euclidean space is compact precisely when it is closed (contains all its limit points) and bounded (can be contained within some large sphere).
Transcribed Image Text:**Theorem 6.20 (Heine-Borel Theorem).** Let \( A \) be a subset of \( \mathbb{R}^n \) with the standard topology. Then \( A \) is compact if and only if \( A \) is closed and bounded. This theorem is a key result in real analysis and topology, connecting the concepts of compactness, closed sets, and boundedness in Euclidean spaces. It establishes that a subset of a Euclidean space is compact precisely when it is closed (contains all its limit points) and bounded (can be contained within some large sphere).
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