Analogues of the Fundamental Isomorphism Theorems in Topology We know that the first fundamental isomorphism theorem for groups has somewhat of a "natural" analog in topology, specifically the one that deals with the quotient space being homeomorphic to the image of identification maps. 
This might be a bit general or vague but I was wondering if there are analogs of the 2nd and 3rd isomorphism theorems in some branch of algebraic topology. I think this also will cross into category theory but just the same, I don't know exactly where to start looking for such analogs.
 A: The first isomorphism theorem is false for topological spaces, in the following sense. 
In many categories arbitrary morphisms $f : x \to y$ have what's called a (regular) image-coimage factorization
$$x \to \text{coim}(f) \to \text{im}(f) \to y$$
where the first morphism is an epimorphism and the last morphism is a monomorphism. A very abstract statement of the first isomorphism theorem is that it means the middle arrow is always an isomorphism. In groups, $\text{im}(f)$ can more or less be identified with the set-theoretic image of a homomorphism in the usual sense, whereas $\text{coim}(f)$ is the quotient of $x$ by the kernel of $f$, and so we recover the usual first isomorphism theorem this way.
For topological spaces, if $f : X \to Y$ is a continuous map, $\text{im}(f)$ is the set-theoretic image of $f$ topologized with the subspace topology from $Y$, and $\text{coim}(f)$ is the set-theoretic image of $f$ topologized with the quotient topology from $X$. These do not agree in general, although they do agree if $X$ and $Y$ are both compact Hausdorff. 
