Let X,Y be two spaces, A a closed subset of X, f:A--->Y a continuous map. We denote by [itex]X\cup_fY[/itex] the quotient space of the disjoint union [itex]X\oplus{Y}[/itex] by the equivalence relation ~ generated by [itex]a ~ f(a)[/itex] for all a in A. This space is called teh attachment of X with Y along A via f.(adsbygoogle = window.adsbygoogle || []).push({});

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].

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Topology - quotient spaces

**Physics Forums | Science Articles, Homework Help, Discussion**