This is a very simple question involving basic definitions. I want to prove that if $G$ is a topological group, left multiplication $f_a\colon g\mapsto ag$ is a homeomorphism of $G$. Clearly, this map is bijective and it sufficies to show its continuity.
To prove continuity, if $U(ag)$ is an open nhbd of $ag\in G$, there are open nhbds $V(a)$, $W(g)$ s.t. $V(a)W(g)\subseteq U(ag)$, from which I get $aW(g)\subseteq U(ag)$.
Here I stuck. Can I conclude it is open? If yes, why?
Thank you in advance for your help.