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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?

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You will certainly find interesting Counterexamples in topology by L.A.Steen and J.A.Seebach, Jr. –  Pietro Majer Aug 30 '10 at 21:52
Years ago I spent some time thinking about ultrafilters. Recall that a filter on 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 is proper if it does not contain the empty set, and an ultrafilter is a maximal proper filter. Given a filter on $X$ and a topological space $Y$, a limit of 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 3:31
(continuation) $X = \mathbb N$ and picking the cofinite 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 3:37
(continuation) back to an element of the ultrafilter. The reason I bring this up is that I think it illustrates some big differences between sequential and limit-point compactness. The reason I leave it as an extended comment is that it really is not an answer to your question. (The correct answer to your question being the one Pietro Majer has already given, namely that Counterexamples in topology is an excellent book, and almost certainly contains what you're looking for.) –  Theo Johnson-Freyd Aug 31 '10 at 3:39
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2 Answers

up vote 3 down vote accepted

I don't have Munkres' book. So I don't know what is done there. You should probably consult the "Counterexamples in Topology" as mentioned above.
My favourite book for questions of this type is "General Topology" by Ryszard Engelking. It has a diagram in the back with interrelations between different properties of topological spaces. It has all the notions you are interested in, except the last two. Local compactness and various additional versions of paracompactness are also discussed in the book. I assume, but haven't checked in every case, that the relations in the diagram are proved in the book. The book also contains plenty of examples, showing that certain implications do not hold, sometimes in the form of exercises with hints.

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You should get Munkres' First Course in Topology and read it if you have not done already. This is not the question for Mathoverflow. It is actually a fair general topology homework assignment for a solid undergraduate student.

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Surely so for some of the notions (especially the implications I stated at the end of the post), but most of them are not even defined in Munkres book. –  Bruno Stonek Aug 30 '10 at 22:55
But why bother learning any one of these notions unless you actually need it? –  Deane Yang Aug 30 '10 at 23:07
I think all but the last three ones are either important or natural to formulate, so it is interesting per se to ask oneself what kind of relationship do they bear. –  Bruno Stonek Aug 30 '10 at 23:10
For me it seems that although it is, up to some extent, a basic general topology question, it is a very good question. Munkres' book doesn't provide you with all the definitions given by Bruno, let alone all their interrelations. One comes across these definitions every now and then, and their relations and relative properties can be useful. The counter examples book can give you some intuition and place to start with but the question won't be answered completely there. –  Dror Atariah Aug 31 '10 at 4:54
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