Combining the axioms of a topological group

Question

According to Wikipedia, a topological group $G$ is a topological space and a group, such that the functions $$(x,y) \mapsto x\cdot y\\x\mapsto x^{-1} $$are continuous. Is the single requirement that $$(x,y) \mapsto x \cdot y^{-1} $$ is continuous sufficient as well? I know that "combining the axioms" works in the case of vector spaces, where closure under scalar multiplication and closure under addition can be combined into closure under linear combinations - however, I don't know if the same holds here.

Thanks!

Answer 1

I'll get you started: assuming your hypothesis, if we plug in $x=1$, we get that

$$y \to y^{-1}$$

is continuous.