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The statement of my problem:

If $f:(X,A)\to(Y,B)$ is a homotopy equivalence of pairs, then so is the induced map $\hat{f}:(X/A, *)\to (Y/B, *)$.

I'm confused on what the star stands for? I think with that I can infer how the induced map works.

Regarding comments:

So If I have the quotient map $q:X\to X/A$, $q(A) =\{A\}$, and then my induced map can be written

$\hat{f}:(X/A, \{A\})\to (Y/B, *)$? Would it make sense to replace the second star by $\{B\}$?

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With some confidence: The first star stands for the image of $A$ in the quotient $X/A$, which is a point. –  Dylan Moreland Feb 2 '12 at 20:06
    
I made some edits to test whether I understand your answer. –  Kyle Schlitt Feb 2 '12 at 20:16
    
What you say is correct. I didn't mention $B$ out of laziness, really. –  Dylan Moreland Feb 2 '12 at 20:19
    
OK. This is intuitive. Thanks! –  Kyle Schlitt Feb 2 '12 at 20:21

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