Over at the nForum, we've been discussing sequential compactness. The discussion led me to realise that I naively assumed that nets were simply Big Sequences, and that I could make a reasonable guess at how nets would behave by thinking of them as such.

Not so. The crucial point, that I hadn't realised, was that subnets are not sub-nets in the way that subsequences are sub-sequences.

Where this came to light was in a discussion of the relationship between compactness and sequential compactness. Compactness can be expressed as:

Every net has a convergent subnet.

Sequential compactness as:

Every sequence has a convergent subsequence.

So, in my naivety, I assumed that compactness implied sequential compactness since I could take a sequence, think of it as a net, find a convergent subnet, and - ta-da - there's my convergent subsequence. The error, as Mike Shulman pointed out, is that not every subnet of a sequence is a subsequence.

And, indeed, there is a space that is compact but not sequentially compact. Writing $I = [0,1]$ then $I^I$ is compact but not sequentially compact. In particular, it is possible to find a sequence that has no convergent subsequence (the argument is a variant of Cantor's diagonal theorem) but that has plenty of cluster points and thus plenty of convergent subnets.

But the compactness of $I^I$ seems to require a Big Axiom (not quite the axiom of choice, or so I'm led to believe since $I$ is Hausdorff, but almost). I say "seems to" since I'm not an expert and there may be a way to prove that this specific space, $I^I$, is compact with only the basic axioms of ZF.

That's basically my question, except that I'm a topologist so I'm more interested in the implications for topological stuff than in the exact relationship between the Axiom of Choice and Tychanoff's theorem (and since I can just read the nLab page to learn that!). So, without further ado, here's the question:

Is "Compactness => Sequential Compactness" consistent with ZF?

This could be answered by a topologist since all it would require to show that this isn't so would be an example of a space that was compact but not sequentially compact and such that proving that didn't require any Big Axioms.

References:- nLab pages: sequential compactness (has more details on the above example), nets (contains the crucial definition of a subnet), Tychonoff's theorem (contains a discussion of the axiomatic strength of this theorem)
- nForum discussion: sequential compactness

Counterexamples In Topologyto see if the matter is resolved there? – Pete L. Clark May 12 '10 at 23:58