Past exam of the metric spaces and complex analysis course of the University of Oxford / 2015 / 1 / a

The Heine-Borel theorem says that a subset $K$ of finite-dimensional Euclidean space is compact exactly when it is both closed and bounded:Here compactness refers to the subspace topology induced by the usual Euclidean metric, and closedness and boundedness are measured in the ambient Euclidean space. This characterization does not hold for arbitrary metric spaces.