- #1

- 175

- 0

i) If A is a strong deformation retract of X, show that Y is a deformation retract of [itex]X\cup_fY[/itex].

ii) The map f is extendable to a continuous map F:X-->Y if and only if Y is a retract of [itex]X\cup_fY[/itex].

I'm really stuck on this question. If A is a strong deformation retract of X, then we have some homotopy [itex]H:X\times[0,1]\rightarrowX[/itex] where for all x in X, H(x,0) = x. H(x,1) is in A. and for all b in A and all t in [0,1], H(a,t) = a.

But what has that got to do with [itex]X\cup_fY[/itex]? :yuck:

Are the members of [itex]X\cup_fY[/itex] the equivalence classes where for a~b if one of: a = b, f(a)=f(b), a is in A and b=f(u) in Y.

So we want to come up with a map [itex]\varphi:X\cup_fY\rightarrow{Y}[/itex] where for all y in Y, [itex]\varphi(y) = y[/itex].