Open mappings and continuous functions? Are these interrelated? In Urysohn's Metrization Theorem, at some point we define a function $F : X \rightarrow H$ from the space X into Hilbert space $H$. Whereupon we need to show that $F$ is an embedding. To show this, it apparently suffices to show that $F$ is * one-to-one * continuous * an open mapping. I don't see how the last two differ? Continuity is proven with open sets, so is an open mapping?

A map is continuous if _preimage of open sets are open sets_ and it is open if _direct image of open set are open sets_.

More precisely, $f:X\to Y$ is **continuous** if $f^{-1}(V)\subset X$ is open for any open set $V\subset Y$.

It is open if $f(U)\subset Y$ is **open** for any open set $U\subset X$.