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Mathematics

Limit Points and Compactness in Metric Spaces

Quick fact

In a metric space, a compact set is one where every infinite subset has a limit point within the set—this is the Bolzano–Weierstrass property, and it is equivalent to the usual open-cover definition of compactness in metric spaces.

Why this is interesting

Imagine a crowd of points gathering closer and closer to a single spot—never quite arriving, but always approaching. What does that tell us about the entire set, and when can we be sure that such a gathering place must exist?