$(x,y)\to xy$ continuous but $x\to x^{-1}$ not

Question

In the definition of topological groups we impose both $(x,y)\to xy$ and $x\to x^{-1}$ to be continuous.

However, I cannot find an example where the first condition holds but the second fails.

Is the second one redundant?

Thanks!

Answer 1

Take $\mathbb{Z}$ with the usual group operation and topology given by the open sets $(n, \infty), n \in \mathbb{Z}$ (together with the empty set and the entire space). The group operation is continuous since the preimage of $(n, \infty)$ is a union of the open sets $(a, \infty) \times (b, \infty)$ where $a + b = n$, but inversion is not since the preimage of $(n, \infty)$ is $(-\infty, -n)$ which is not open.

Answer 2

Sorgenfrey Line is another example of paratopological group. See: Topological group and related structure, Book by Arhangel'skii and Tkachenko Page 13 example 1.2.1.