Topological groups, is continuity a requirement or consequence of this definition?

Question

On the Wikipedia page for topological groups, it is stated as a requirement that the binary operation and the inverse operation is continuous.

In Munkres he makes this definition:

A topological group G is a group that is also a topological space statisfying the $T_1$ axiom, such that the map $G\times G$ into $G$ setnding $x\times y$ into $xy$ , and the map of $G$ into $G$ sending $x$ to $x^{-1}$, are continuous maps.

I am a little unsure about the word such "such". Does he also mean that you require the functions to be continuous, or can you with the $T_1$ axiom in fact show that they must be continuous? I tried proving it, but came nowhere.

Answer 1

He means that the requirements are:

1. The topology is $T_1$.
2. The operation $(x,y)\to xy$ is continuous from $G\times G$ with it's product topology to $G$.
3. The operation $x\to x^{-1}$ is continuous from $G$ to $G$.

You cannot prove 2. and 3. from just assuming 1.

In another classic, namely, "Topological groups" by Pontryagin, there is no requirement that the topology be $T_1$.