Is the inclusion always an homeomorphism? If not, which conditions must we have for such an assertion to be true?
What's the relationship between the image of the inclusion having the subspace topology, and the inclusion being a homeomorphism?
Any help would be appreciated
An inclusion map is any function $j:A\to B$, where $A$ is a subset of $B$ and $j(a)=a$ for all $a\in A$. (Note that $A$ and $B$ need not have any particular structure in this case.)
A homeomorphism is any map $u:A\to B$, where $A$ and $B$ are topological spaces, $u$ is bijective, and both $u$ and $u^{-1}$ are continuous.
The only kind of map which is both an inclusion and a homeomorphism, therefore, is the identity map from a topological space onto itself.
EDIT: However, it is worth noting that every inclusion map $f:A\to B$, where $B$ is a topological space and $A$ is a subspace (endowed with the subspace topology) is a homeomorphism when viewed as a map $f:A\to f(A)$ (where $f(A)$ is also endowed with the subspace topology).
EDIT #2: Also, in the context of topological spaces, if $C$ and $B$ are topological spaces, $A$ is a subspace of $B$ (endowed with the subspace topology), and $u:C\to A$ is homeomorphism, then we may "abuse notation" and write that $u$ is an inclusion map when viewed as a function from $C$ into $B$.