Given a topological space X one can define several notion of compactness:

X is **compact** if every open cover has a finite subcover.

X is **sequentially compact** if every sequence has a convergent subsequence.

X is **limit point compact** (or Bolzano-Weierstrass) if every infinite set has an accumulation point.

X is **countably compact** if every countable open cover has a finite subcover.

X is **σ-compact** if it is the union of countably many compact subspaces.

X is **pseudocompact** if its if its image under any continuous function to $\mathbb{R}$ is
bounded.

X is **paracompact** if every open cover admits an open locally finite refinement (i.e. every point of X has a neighborhood small enough to intersect only finitely many members of the cover).

X is **metacompact** if every open cover admits a point finite open refinement (i.e. if every point of X is in only finitely many members of the refinement).

X is **orthocompact** if every open cover has an interior preserving open refinement (i.e. given an open cover there is a open subcover such that at any point, the intersection of all open sets in the subcover containing that point is also open).

X is **mesocompact** if every open cover has a compact-finite open refinement (i.e. given any open cover, we can find an open refinement such that every compact set is contained in finitely many members of the refinement).

So, there are quite a few notions of compactness (there are surely more than those I quoted up here). The question is: where are these definitions systematically studied? What I'm interested in particular is knowing when does one imply the other, when does it not (examples), &c.

I can fully answer the question for the first three notions:

Compact and first-countable --> Sequentially compact.

Sequentially compact and second-countable --> Compact.

Sequentially compact --> Limit-point compact.

Limit point compact, first-countable and $T_1$ --> Sequentially compact.

but I'm absolutely ignorant about the other cases. Has this been systematically studied somewhere? If so, where?

Counterexamples in topologyby L.A.Steen and J.A.Seebach, Jr. – Pietro Majer Aug 30 '10 at 21:52filteron a set $X$ is a collection of subsets of $X$ such that if $A,B \subseteq X$ are in the filter, so is $A\cap B$, and if $A\supseteq B \supseteq X$ and $A$ is in the filter, then so is $B$. A filter isproperif it does not contain the empty set, and anultrafilteris a maximal proper filter. Given a filter on $X$ and a topological space $Y$, alimitof a function $f : X\to Y$ is a point $y\in Y$ such that every open containing $y$ pulls back to an element of the filter. For example, letting (continued) – Theo Johnson-Freyd Aug 31 '10 at 3:31cofinite filter(a subset of $\mathbb N$ is in the filter if its complement is finite) returns the usual Cauchy definition of the limit of a sequence. If $Y$ is compact (and Hausdorff) and we pick an ultrafilter on $X$, then every function has a unique limit (necessarily, if the ultrafilter contains the cofinite filter, to an accumulation point). On the other hand, my adviser for the project gave an example of a sequence in $[0,1]$ and an ultrafilter on $\mathbb N$ so that no convergent (in the cofinite filter) subsequence pulled (continued) – Theo Johnson-Freyd Aug 31 '10 at 3:37Counterexamples in topologyis an excellent book, and almost certainly contains what you're looking for.) – Theo Johnson-Freyd Aug 31 '10 at 3:39