CHAPTER 2 - SPACES AND THEIR PATHS

Summary

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 X_αbe subsets of X such that X-∪X_α, and let f_α:X_α → Y be functions such that f_α|X_α∩X_β – f_β|X_α∩X_βfor all α and β. Then there is a unique function f:X→Y such that f|X_α – f_αfor all α.

(ii) Let the conditions of (i) hold, and let X and Y be topological spaces. Suppose that each f_αis continuous (when X_αis given the subspace topology). Suppose either that there are only finitely many sets X_αeach of which is a closed subspace of X or that each X_αis 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∈X_α and y-xf_α). Since X-⊃X_α, 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∈X_α, y - xf_α, and x∈X_β, z - xf_β. Since fsub>α - f_β on fsub>α ∩ f_β, by hypothesis, it follows that y - z.