# Trouble understanding topological groups.

I understand that a topological group is a group $$G$$ endowed with a topology $$\tau$$ on $$G$$ such that addition and inverse are continuous on $$\tau$$.

Now, the definition of continuity is that for all $$U\in\tau$$, $$f^{-1}(U)\in\tau$$. But in this case $$+^{-1}(U)$$ is a subset of $$G\times G$$ because the domain of $$+$$ is $$G\times G$$, and we endowed $$G$$ with a topology, not $$G\times G$$. So, where am I messing up?

• $G\times G$ has the product topology – Max Feb 14 at 17:55
• I love this question. – Ben Blum-Smith Feb 14 at 18:11
• @BenBlum-Smith why? – Alex Kruckman Feb 15 at 3:47
• @AlexKruckman - I recognize an earnest desire to understand. – Ben Blum-Smith Feb 16 at 4:55
• @BenBlum-Smith That's a good reason! – Alex Kruckman Feb 16 at 5:22

The topology of $$G\times G$$ is the product topology, which is the topology of the unions of sets of the form $$A\times B$$, with $$A,B\in\tau$$.
• And in brief, continuity of $+$ in general unfolds to: whenever $x+y \in U \in \tau$, there exist $V,W$ with $x \in V \in \tau, y \in W \in \tau$, and $V+W = \{ x' + y' \mid x'\in V, y' \in W \} \subseteq U$. – Daniel Schepler Feb 14 at 19:00