Problem 1: A continuous injective map that is either open or closed is a topological embedding.
Solution:Without loss of generality suppose $f:X\rightarrow Y$ is a continuous injective open map. Then $f: X\rightarrow f(X)$ is a continuous bijection. To show that it is a homeomorphism, it suffices to show that $f$ onto its image is open. Let $U$ be open in $X$ so by assumption, $f(U)$ is an open subset of $Y$. Since $f(U)\subseteq f(X)$, $f(U)= f(U)\cap f(X)$, which is open in $f(X)$. Hence $f$ onto its image is a homeomorphism. Thus $f$ is a topological embedding.
Problem 2:
A surjective topological embedding is a homeomorphism
Solution:Suppose $f:X\rightarrow Y$ is a surjective topological embedding, so $f:X\rightarrow f(X)$ is a homeomorphism, but $f(X)=Y$ since $f$ is surjective, so $f:X\rightarrow Y$ is a homeomorphism.
Are the solutions correct?
Best Answer
For problem 1, your argument is correct but it wouldn't hurt to include a proof of the fact that a continuous bijective open map is a homeomorphism, as this is not entirely trivial.
For problem 2, your argument is correct (it is a rather trivial question to ask, though).