In John Lee's Introduction to Topological Manifolds, the following theorems are proven:
- For all spaces:
- Compactness $\Rightarrow$ limit point compactness
- For first countable Hausdorff spaces:
- Limit point compactness $\Rightarrow$ sequential compactness
- For metric spaces and second countable Hausdorff spaces:
- Compactness $\Leftrightarrow$ limit point compactness $\Leftrightarrow$ sequential compactness
Can we strengthen this list so that, given a set of compactness equivalences, the topological properties of a space are precisely known?
In other words, can we fill in the following blanks:
"Compactness $\Leftrightarrow$ limit point compactness" holds on (and only on) ___ spaces."
"Compactness $\Leftrightarrow$ sequential compactness" holds on (and only on) ___ spaces."
"Limit point compactness $\Leftrightarrow$ sequential compactness" holds on (and only on) ___ spaces."