Sequential Compactness

Topology · Axiom Academy

Unit 5 - Compactness - Sequences and compactness Step 1: Sequential Compactness Definition Sequential compactness provides a powerful way to understand compact spaces through the behavior of sequences. Instead of working with open covers, we ask: what happens to sequences in this space? A topological space is sequentially compact if every sequence in has a convergent subsequence that converges to a point in . Step 2: Visualizing Sequences and Subsequences Let's visualize what it means for a sequence to have a convergent subsequence. Watch how we can extract a subsequence from any sequence in a sequentially compact space. Consider a sequence in . Even if the sequence doesn't converge, we can always find a subsequence that does! Step 3: The Bolzano-Weierstrass Theorem One of the most important results connecting boundedness and sequential compactness is the Bolzano-Weierstrass theorem. Every bounded sequence in has a convergent subsequence. Equivalently: Every bounded infinite subset of has a limit point. Step 4: Sequential Compactness in Metric Spaces In metric spaces, sequential compactness has a beautiful characterization. Let's explore what properties guarantee that a metric space is sequentially compact. A metric space is sequentially compact if and only if it is complete and totally bounded . A metric space is totally bounded if for every , there exists a finite set of points such that . In other words: we can cover with finitely many balls of any given radius.

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