Homotopic maps are close

From Topospaces
Jump to: navigation, search

Statement

Suppose X is a compact space and (Y,d) is a metric space. Suppose f and g are homotopic maps from X to Y. Then, there exists an \epsilon > 0 and a sequence of maps f=f_0,f_1,f_2,\ldots,f_n = g such that for every x \in X:

d(f_i(x),f_{i-i}(x)) < \epsilon

A related fact is that close maps are homotopic: the condition of compactness is now on Y instead of on X.