We let $f$ be a continuous function $f:X\rightarrow Y$ where $X$, $Y$ are two topological spaces. If $g:T\rightarrow Y$ is a continuous function s.t for every $t\in T$ there exists $x\in X$ s.t $g(t)=f(x)$ then there exists a unique continuous function $\theta:T\rightarrow X$ s.t $f\circ\theta=g$.
I want to prove that such $f$ is an embedding, meaning $f:X\rightarrow \text{Im}f$ is a homeomorphism. Proving $f$ is injective wasn't too hard, but proving that the inverse is continuous proved to be quite difficult. Any help would be appreciated.
Best Answer
Initial thoughts:
There is hardly any choice at all in defining $\theta$, by $f \circ \theta = g$, we must define $\theta(t)$ $(t$ given so $g(t)$ as well) as some $x$ promised by your property (so there is at least one), or any other $x \in X$ that obeys $f(x)=g(t)$. The fact that $\theta$ must be continuous should "force" the right $x$, presumably.