1. Continuity This document collects statements of useful facts that give helpful shortcuts for proving continuity of maps. Sometimes, though, you just have to check the definition! First, the basic definition: Definition 1.1. A function f : X ! Y between topological spaces is continuous if, for every open subset U of Y , f −1(U) is an open subset of X. Or equivalently, in terms of closed sets: Proposition 1.2. A function f : X ! Y between topological spaces is continuous if and only if, for every closed subset Z of Y , f −1(Z) is a closed subset of X. 1.1. Using a basis. Proposition 1.3. Let f : X ! Y be a function between topological spaces, and let B be a basis for the topology on Y . Then f is continuous if and only if f −1(U) is open in X for every U 2 B. 1.2. Compositions. Proposition 1.4. If f : X ! Y and g : Y ! Z are continuous maps between topological spaces, then so is g ◦ f : X ! Z. 1.3. Patching continuous maps. If the domain of a function is covered by open subsets, it suffices to check continuity on each of those subsets: Proposition 1.5. If f : X ! Y is a function between topological spaces and X is a (possibly S infinite) union i Vi of open subsets, then f is continuous if and only if its restriction to every Vi is a continuous map Vi ! Y . The analogous statement for closed subsets works if there are only finitely many: Proposition 1.6. If f : X ! Y is a function between topological spaces and X is a finite S union i Zi of closed subsets, then f is continuous if and only if its restriction to every Zi is a continuous map Zi ! Y . 1.4. Inverses of continuous bijections. In general the inverse of a continuous bijection need not be continuous. However: Proposition 1.7. If f : C ! H is a continuous bijection from a compact space to a Hausdorff space, then f −1 is also continuous (and so f is a homeomorphism). 1.5. Subspaces. It is relatively easy to understand whether a map to a subspace is contin- uous. Proposition 1.8. Let W and Y be topological spaces, let X ⊆ Y , considered as a topological space with the subspace topology, and let i : X ! Y be the inclusion map. (1) i is continuous. (2) A map f : W ! X is continuous if and only if i ◦ f : W ! Y is continuous. [In other words, a map to X is continuous iff it is continuous when considered as a map to the larger space Y .] 1 2 1.6. Products. It is relatively easy to understand whether a map to a product is continuous. Proposition 1.9. Let W , X and Y be topological spaces. Consider X × Y as a topological space with the product topology, and let px : X × Y ! X and pY : X × Y ! Y be the maps projecting onto the first and second coordinates. (1) pX and pY are continuous. (2) A map f : W ! X × Y is continuous if and only if pX ◦ f and pY ◦ f are continuous. [In other words, if f(w) = (fX (w); fY (w)), then f is continuous iff both fX : W ! X and fY : W ! Y are continuous.] 1.7. Disjoint unions. It's easy to understand whether a map from a disjoint union is continuous. Proposition 1.10. Let X, Y and W be topological spaces, and let X t Y be the disjoint union of X and Y . A map f : X t Y ! W is continuous if and only if its restrictions to X and Y are both continuous. 1.8. Quotient spaces. It is relatively easy to understand whether a map from a quotient space is continuous. Proposition 1.11. Let X and W be topological spaces, let ∼ be an equivalence relation on X. Let X= ∼ be the quotient space and q : X ! X= ∼ the quotient map q(x) = [x]. (1) q is continuous. (2) A map f : X= ∼→ W is continuous if and only if f ◦ q : X ! W is continuous..