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I have some conditional questions. First, in the category of topological pairs (as used in cohomology theories), are these only pairs of the form $(X,A)$ with $A \subset X$?

If the answer to this question is yes, then my next question is: is a morphism $(X,A) \to (Y,B)$ a map $f:X \to Y$ with $f(A) \subset B$ or pairs of maps $f_1: X \to Y$ and $f_2: A \to B$?

Thanks!

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    The answer to your first question is yes. For the second question, the answer is the first.2011-02-06
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    Thanks Akhil. I asked because I've seen the Thom isomorphism phrased in terms of $H^*(E, E - B)$ and $H^*(D(E), S(E))$, where $E \to B$ is a vector bundle, $D(E)$ is the unit disk and $S(E)$ the sphere. I wanted the pairs $(E,E-B)$ and $(D(E), S(E))$ to be homotopic. But I'm guessing the long exact sequence in cohomology and the five-lemma give that their cohomology groups are naturally isomorphic.2011-02-06

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