If $X$ is a metrizable topological space where the only convergent sequences are eventually constant sequences, then $X$ must be a discrete space. But I'm interested in whether something stronger is true.
Let $X$ be a uniformizable topological space, i.e. a completely regular space, where the only convergent sequences are eventually constant sequences, must $X$ necessarily be a discrete space?
If it's not true, does anyone know of a counterexample?
On
Eric has given an elementary example, as an addition and to see that such spaces actually occur in "real life" (in as far as pure maths can be), consider the space $\omega^\ast$, the remainder of $\omega$ in its Čech-Stone compactification. This is compact Hausdorff (so certainly completely regular; it's also zero-dimensional like Eric's example) and has no non-trivial convergent sequences. (It's also the Stone space of $\mathscr{P}(\omega)/\text{Fin}$, if you know such theory.)
No. For instance, let $X$ be an uncountable set, fix a point $a\in X$, and define a topology on $X$ by saying a set is open iff it either is cocountable or does not contain $a$. Then $X$ is completely regular (it has a basis of clopen sets, namely the singletons other than $a$ and the cocountable sets containing $a$), and every convergent sequence in $X$ is eventually constant (since every cocountable set is open).