Compactness Summary

Topology · Axiom Academy

Unit 5 - Complete review of compactness and its key properties Overview: What is Compactness? Compactness is a fundamental topological property that generalizes the notion of closed and bounded sets in Euclidean spaces. It captures the idea that a space is "small" in a topological sense, even if it's infinite. Compact spaces have many remarkable properties that make them central to analysis and topology. A collection of open sets is an open cover of a set if . A topological space is compact if every open cover of has a finite subcover. That is, if is an open cover of , then there exist finitely many sets such that . Visual Recap: Open Cover & Finite Subcover An open cover (many sets) can be reduced to a finite subcover (highlighted sets) that still covers the space A subset of is compact if and only if it is closed and bounded . Closed intervals like are compact Open intervals like are NOT compact (not closed) Unbounded sets like are NOT compact A closed subset of a compact space is compact. Formula: If is compact and is closed in , then is compact. The continuous image of a compact space is compact. If is continuous and is compact, then is compact. A space is compact iff every collection of closed sets with FIP has nonempty intersection. FIP: Every finite subcollection has nonempty intersection. In metric spaces, compactness is equivalent to sequential compactness. Every sequence has a convergent subsequence (Bolzano-Weierstrass). Compactness and Continuous Functions

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