1
$\begingroup$

A topological group is one in which the group operations (the multiplication and inverse) are continuous, or equivalently as a group object in $\mathbf{Top}$.

They are uniformisable and hence are completely regular & $R_0$.

Can anything similar i.e. with respect to the separation axioms be said about topological monoids?

$\endgroup$
2
$\begingroup$

The space $X=(\Bbb N,+,0)$ with the topology generated by the sets $$A_n=\{0,1,...,n\}$$ is a topological monoid. If $m,n$ are naturals, then the smallest neighborhood of $m+n$ is $A_{m+n}$, and it contains the sum $A_m+A_n$, so addition is continuous.
$X$ is $T_0$, but not $T_1$, so it isn't uniformizable.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.