Prove that compactness is a topological property SolutionSup

Prove that compactness is a topological property.

Solution

Suppose that X and Y are topological spaces, and f : X Y is a homeomorphism. That is, f : X Y is a bijection, f and f ¹ : Y X are continuous.

(i) Compactness is a topological property:

This is to show that if X is compact, then Y is also.

Suppose that X is compact. We want to show that Y is also compact.

Our strategy is to apply the definition, and show that every infinite sequence of points in Y has a limit point in Y . Let {yi} _i=1 be an infinite sequence of points in Y .

Let x_i = f ¹ (yi), i = 1, 2, ... Then {xi}_i=1 is an infinite sequence of points in X.

Since X is compact, {xi}_i=1 has a limit point a in X.

Since f : X Y is continuous, f(a) is a limit point of {f(xi)}_i=1 = {yi}_i =1.

Since f(a) Y , the sequence {yi}_i=1 has a limit point in Y .

By the definition of compact sets, Y is compact.