1. Continuity This Document Collects Statements of Useful Facts That Give Helpful Shortcuts for Proving Continuity of Maps. Some

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 ∈ 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 continuous. 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.