Sequential compactness and limit point compactness

Let $X$ be a topological space (In particular, $X$ is not metric space nor $T_1$ space). Is there any implication between sequential compactness and limit point compactness?

I found many posts related to this topic but I think all the question assumes 'metrizable space' or '$T_1$ space' (in particular, every neighborhood of a limit point of some subset $A\subset X$ contains infinitely many points of $A$). In general setting, is there any implication between these two notions?

Edit. I just realized that sequential compact $\Rightarrow$ countably compact $\Rightarrow$ limit point compact. I bet limit point compact $\not\Rightarrow$ sequential compact. Any counterexample for this?

Solution 1:

The “fat double” of $\Bbb N$ is limit point compact but not sequentially compact nor countably compact. For a $T_1$ space, with some work, we can show that a limit point compact space is in fact countably compact.

But it’s clear that a sequentially compact space $X$ is always strongly limit compact: if $A$ is infinite we can find (by ACC) an injective sequence $(a_n)$ in $A$ and if $p$ is such that $a_{n_k} \to p$ for some subsequence, any neighbourhood of $p$ contains a tail of the subsequence, so in particular $p$ is an $\omega$-limit point of $A$ and $X$ is is even countably compact.

Finally, $[0,1]^{\Bbb R}$ is (countably) compact but not sequentially compact.