It can be shown, for metric spaces, that the notions of compactness, limit-point compactness and sequential compactness are equivalent, yet these results can be made more general. In particular:
$$\text{compact} \Rightarrow \text{limit-point compact}$$ $$\text{compact} + \text{first-countable} \Rightarrow \text{sequentially compact}$$ $$\text{sequentially compact} \Rightarrow \text{limit-point compact}$$ $$\text{sequentially compact} + \text{second-countable} \Rightarrow \text{compact}$$
I am wondering:
- Which axioms of countability/separation are specifically needed for limit-point compactness to imply compactness?
What I have found so far is a proof that $$T_1 + \text{limit-point compact} \Rightarrow \text{countably compact}.$$
- Which axioms of countability/separation are specifically needed for limit-point compactness to imply sequential compactness?
I believe I remember -although I cannot seem to find where I read it- that $$\text{first-countability} + T_1 + \text{limit-point compact} \Rightarrow \text{sequentially compact}.$$ What I have been able to find are examples of first-countble limit-point compact spaces that are not sequentially compact.
Whatever the answer to these questions are, I'd appreciate if a (link to a) proof is included (:
