Let me start by recalling some basic definitions (just for the sake of avoiding misunderstandings due to the vocabulary of the post).

Basically following [some ideas of W. Lawvere][1] (but not his terminology), we let a _semimetric_ on a set $X$ be a function $d: X \times X \to [0,\infty]$ such that $d(x,x) = 0$ and $d(x,z) \le d(x,y) + d(y,z)$ for all $x,y,z \in X$. Exactly as with "classical" metrics, $d$ induces a topology on $X$, which we call the _canonical topology_ of $d$, having as a base the sets $\{y \in X: d(x,y) < r\}$ as $x$ ranges over $X$ and $r$ over $\mathbf R^+$.

Next, we take a _topological monoid_ to be, as usual, a pair $(\mathbb M, \tau)$ consisting of a (multiplicatively written) monoid $\mathbb M = (M, \cdot)$ and a topology $\tau$ on $M$ such that $\cdot$ is continuous [...] in the expected sense.

Finally, if $\mathbb M = (M, \cdot)$ is a monoid and $d$ is a semimetric on $M$, we say that $d$ is: _right subinvariant_ (in $\mathbb M$) if $d(xz,yz) \le d(x,y)$ for all $x,y,z \in M$; _left_ subinvariant (in $\mathbb M$) if it is right subinvariant in the dual of $\mathbb M$; and _subinvariant_ (in $\mathbb M$) if it is both right and left subinvariant.

With all of this in mind, what is known about the following question?

>> Let $(\mathbb M, \tau)$ be a topological monoid, with $\mathbb M = (M, \cdot)$. Does there always exist a left (respectively, right) subinvariant semimetric $d$ on $M$ such that $\tau$ is the canonical topology of $d$? And what about a subinvariant semimetric?

Thanks in advance for any possible pointer.


  [1]: http://www.tac.mta.ca/tac/reprints/articles/1/tr1.pdf