# 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?

• @yili I don't think so. It states the relation between those two in metric space. Commented Nov 27, 2021 at 11:18

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.
• I've never heard that 'fat double of $\Bbb N$' space and I can't find that space is google. Could you explain what that space is? It would be nice if you verify why such space is limit point compact but not sequentially compact. Commented Nov 27, 2021 at 13:40
• @love_sodam it’s $\Bbb N$ in the discrete topology times the indiscrete two point space $\{0,1\}$ in the product topology. Think about why it has the properties I mentioned. Commented Nov 27, 2021 at 13:43