A subspace A is deformation retract of X, if there exists a retraction r: X A...

Transcribed Image Text:

A subspace A is deformation retract of X, if there exists a retraction r: X A homotopic to the identity map ix on X. (a) Show that if A is a deformation retract of X, then 7(X, Z) = (A. To) for any zo A. [Hint: Use the fact that if f.g: (X, To) (Y.yo) and f g (via H) such that H(o, t) = y0, Vt, then f. = 9..] (b) Show that S is a deformation retract of Rn+1\ {0} for n 1. (c) Show that R cannot be homeomorphic to R" for n > 2. A subspace A is deformation retract of X, if there exists a retraction r: X A homotopic to the identity map ix on X. (a) Show that if A is a deformation retract of X, then 7(X, Z) = (A. To) for any zo A. [Hint: Use the fact that if f.g: (X, To) (Y.yo) and f g (via H) such that H(o, t) = y0, Vt, then f. = 9..] (b) Show that S is a deformation retract of Rn+1\ {0} for n 1. (c) Show that R cannot be homeomorphic to R" for n > 2.

Answer rating: 100% (QA)

The detailed ... View the full answer