A subset \( A \subset \mathbb{R}^n \) is compact if and only if \( A \) is closed and bounded.

Proof

“\(\Rightarrow\)”: This follows, since \(\mathbb{R}^n\) is metric space and thus a Hausdorff space, i.e. every compact set is bounded and closed.

“\(\Leftarrow\)”: \( A \) is contained in some cube \([-R, R]^n\). Since \([-R, R]\) is compact [1, 4.39], \([-R, R]^n\) is compact as a product. Because \( A \) is a closed subset of \([-R, R]^n\), it is compact.

References Link to heading

  1. J. Lee, Introduction to Topological Manifolds. New York, NY: Springer New York, 2011. doi:10.1007/978-1-4419-7940-7