SOME POINT-SET TOPOLOGY

We begin with a lemma that will be frequently used, often without explicit mention.

Lemma 1 (Glueing Lemma) (i) Let X and Y be sets, let Xαbe subsets of X such that X-∪Xα, and let fα:Xα → Y be functions such that fα|Xα∩Xβ – fβ|Xα∩Xβfor all α and β. Then there is a unique function f:X→Y such that f|Xα – fαfor all α.

(ii) Let the conditions of (i) hold, and let X and Y be topological spaces. Suppose that each fαis continuous (when Xαis given the subspace topology). Suppose either that there are only finitely many sets Xαeach of which is a closed subspace of X or that each Xαis an open subspace of X. Then f is continuous.

Remark When f is a function from a set x to a set Y and A is a subset of X the notation f\A means the restriction of f to A; that is, the function from A to Y whose value on a ∈ A is fa.

Proof (i) Let S⊆XxY be {(x,y); there is α such that x∈Xα and y-xfα). Since X-⊃Xα, for every x there is at least one y with (x,y)∈S. Suppose that (x,y)∈S and (x,y)∈S. Then there are α and β with x∈Xα, y - xfα, and x∈Xβ, z - xfβ. Since fsub>α - fβ on fsub>α ∩ fβ, by hypothesis, it follows that y - z.