Definition:Compact Space/Euclidean Space

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Definition

Let $\R^n$ denote Euclidean $n$-space.

Let $H \subseteq \R^n$.


Then $H$ is compact in $\R^n$ if and only if $H$ is closed and bounded.


Real Analysis

The same definition applies when $n = 1$, that is, for the real number line:


Let $\R$ be the real number space considered as a topological space under the Euclidean topology.

Let $H \subseteq \R$.


Then $H$ is compact in $\R$ if and only if $H$ is closed and bounded.


Complex Analysis

Definition:Compact Space/Complex Analysis


Also see

  • Results about compact spaces can be found here.


Sources