How does compactness in Euclidean space relate to being closed and bounded?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In the context of Euclidean space, the relationship between compactness and the properties of being closed and bounded is defined by the Heine-Borel theorem.

This theorem establishes that a subset of Euclidean space is compact if and only if it is both closed and bounded. While compactness is defined more broadly in general topological spaces through the existence of finite sub-coverings for any open cover, the specific geometric constraints of Euclidean space allow for this simpler characterization.1

Additionally, compact sets in these spaces satisfy the Bolzano-Weierstrass property, meaning every infinite subset has an accumulation point within the set. These properties are essential in mathematics because continuous functions defined on compact sets are guaranteed to attain both maximum and minimum values.1