Algebraic Topology · Problem 12

The Fundamental Group — Allen Hatcher

Mathematics — Topology · homotopy-equivalence

Problem

Show that a homotopy equivalence

f: X \to Y

induces a bijection between the set of path-components of

X

and the set of path-components of

Y

, and that

f

restricts to a homotopy equivalence from each path-component of

X

to the corresponding path-component of

Y

. Prove also the corresponding statements with components instead of path-components. Deduce that if the components of a space

X

coincide with its path-components, then the same holds for any space

Y

homotopy equivalent to

X