Countable hemi-compact spaces

Question

Apologies for asking about something that ought to be easily googlable but apparently I failed this time.

Is there a countably infinite, completely regular, non-discrete hemi-compact space?

Note that such space does not have convergent sequences as every compact set is finite.

EDIT: I asked a silly question as I had in mind spaces that do no contain infinite compact subsets.

Answer 1

Note that such space does not have convergent sequences as every compact set is finite.

What makes you think so? BTW Every topological space has convergent sequences: trivial sequences, i.e. these that are eventually constant. So I assume you've meant "does not have non-trivial convergent sequences" which still is not true as you can see in the example that follows.

Is there a countably infinite, completely regular, non-discrete hemi-compact space?

Sure, take any convergent sequence $(a_n)\subseteq M$ in a metric space $M$ and put

$$X:=\{a_n\}_{n\in\mathbb{N}}\cup\{\lim(a_n)\}$$

It is compact (and thus hemi-compact) and completely regular (since it is a metric space).

Now if $a_n\neq a_m$ whenever $n\neq m$ (e.g. $a_n=\frac{1}{n}$ in $\mathbb{R}$) then it is countable and not discrete because the limit point is not isolated. Note that $(a_n)$ itself is a non-trivial convergent sequence in $X$.