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 $\pi_n(X,A) \cong \pi_n(X/A)$