Every Closed Subset of a Compact Space is Compact Proof | Summary and Q&A

TL;DR

This video explains how to prove that every closed subset of a compact space is compact.

Key Insights

• 🤗 Compactness means that, for any open covering, it is possible to find a finite subcover.
• 😚 Closed subsets and compact spaces are defined and used in the proof of compactness.
• 😚 The union of a closed subset and its complement can be used as an open covering for the compact space.

Transcript

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Questions & Answers

Q: What does it mean for a set to be compact?

A set is compact if, for any open covering, it is possible to find a finite subcover that still covers the set completely.

Q: How do you prove the compactness of a closed subset?

To prove the compactness of a closed subset, you start by assuming that the subset is closed and that the entire space is compact. Then, you write the overall space as a union of the closed subset and the complement of the closed subset. By constructing a finite subcover using this union, you can demonstrate the compactness of the closed subset.

Q: Why is it important for the closed subset to be closed in the proof of compactness?

The closed subset being closed allows us to write the overall compact space as a union of the closed subset and its complement. This union then becomes an open covering for the closed subset, which is crucial for proving its compactness.

Q: What happens if some elements of the open covering don't cover the closed subset?

If there are elements in the open covering that don't cover the closed subset, you can still include them in the finite subcover. This is because the closed subset is contained within the union of the finite subcover and doesn't intersect the complement set.

Summary & Key Takeaways

• The video discusses the definition of compactness and its implication for closed subsets of a compact space.

• It explains the concept of an open covering and the need to find a finite subcover for compactness.

• The video demonstrates how to use the fact that a subset is closed to write the compact space as a convenient union, enabling the construction of a finite subcover and proving the compactness of the closed subset.