Answer to Question #217675 in Differential Geometry | Topology for Prathibha Rose

Question

State and prove Gluing lemma

Answer 1

Let X,Y be both closed (or both open) subsets of a topological space A such that A=X$\land$ Y, and let B also be a topological space. If f:A$\to$ B is continuous when restricted to both X and Y, then f is continuous.

Proof: if U is a closed subset of B, then f^-1(U)$\land$ X and  f^-1(U)$\land$ Y are both closed since each is the preimage of f when restricted to X and Y respectively, which by assumption are continuous. Then their union,  f^-1(U) is also closed, being a finite union of closed sets.

A similar argument applies when X and Y are both open.