Difference between X/A and G/H I am primarily a student of physics and am trying to self-learn some algebraic topology. I am having some difficulty understanding the differences between the constructions of
$(X,A)$ (Pair of spaces),
$X/A$ (Quotient space),
$G/H$ (Quotient group of topological groups),
$G/H$ (Orbit space where H is viewed as acting on $G$ say by left multiplication)
My questions are as follows:


*

*If $G$ is a topological group and $H$ is a (normal) subgroup then is the quotient group $G/H$ (topologically) the same as $G/H$ viewed as a quotient topological space? If not is there a condition on the topologies or spaces in which they coincide? How does the orbit space $G/H$ differ from these two notions?

*I think I always took for granted that $(X,A)$ was the same as $X/A$ (quotient space) due to excision in homology but now that I am learning some homotopy theory I am not so sure. Is $(X,A)$ ever the same as $X/A$? 

*Under what conditions is $\pi_n(X,A) \cong \pi_n(X/A)$
 A: Let's tackle number $1$ first. $G/H$ viewed as a quotient space in the topological sense is very different from the quotient group.  For example, consider $G=\mathbb R^2$ as an abelian group under addition. Let $H=\mathbb R\times\{0\}$. Then $G/H$ as a topological space is just $\mathbb R^2$ with the $x$-axis shrunk to a point. As a group $G/H$ is homeomorphic $\mathbb R$. So the topological quotient is almost everywhere $2$-dimensional, whereas the group quotient is $1$-dimensional.
For number $2$ you are probably thinking that $H_n(X,A)\cong H_n(X/A)$ in many circumstances, so that roughly "$(X,A)=X/A$". The problem here is that $H_n(X,A)$ is a notational convention for relative homology based on chains $C_n(X,A)=C_n(X)/C_n(A)$. We are not thinking of $(X,A)$ as being itself a space. So really the question doesn't make semantic sense.
As for number $3$, usually one assumes that $A$ is a closed subspace with an open neighborhood that deformation retracts onto $A$. Then your statement is true.
