Compactness

Concepts

• Open cover 10 exercisesAn open cover of a subset A of a topological space X is a family U of open sets U of X whose union contains A.
• Compact space 10 exercisesA topological space X is compact if every open cover U of X has a finite subcover.
• Compact sets in Hausdorff spaces 10 exercisesEvery compact subset K of a Hausdorff space X is closed in X, and every closed subset of a compact space is compact.
• Continuous image of a compact space 10 exercisesIf f:X→ Y is continuous and the space X is compact, then the image f(X) is a compact subspace of Y.
• Heine-Borel theorem 10 exercisesA subset K of ℝ^n with the usual metric is compact if and only if it is closed and bounded.
• Sequential compactness 10 exercisesA metric space X is sequentially compact if every sequence (x_n) in X has a subsequence (x_n_k) converging to a point x∈ X.
• Lebesgue number 10 exercisesA Lebesgue number of an open cover U of a metric space is a δ>0 such that every subset A of diameter less than δ fits inside some…