Let $M$ be a (multiplicative) monoid with a topology $\tau$. I'd like some simple conditions for $(M,\tau)$ to be a topological monoid.
For example, a group $G$ with a topology is a topological group iff
- the "translations" $L_a:x\in G\mapsto ax\in G$ and $R_a:x\in G\mapsto xa\in G$ are continuous,
- the inversion $x\in G\mapsto x^{-1}\in G$ is continuous
- and the product map $(x,y)\in G\times G\mapsto xy\in G$ in continuous at the identities $(e,e)\in G\times G$.
In the case of monoids, if we assume that the translations are open maps, then the analogous conditions to the ones above also imply that the product is continuous. But this is too strong (for example, $[0,1]$ with the operation $x\cdot y=\max(x,y)$ and the usual topology is a topological monoid, but the translations are not open).
Thanks in advance.