Whilst one can study a topological space by itself, it is often useful to probe it with known spaces. A common choice for topological spaces, and in particular metric spaces, is to use the natural numbers, and the 1-point compactification of the natural numbers. This is more traditionally known as studying the topology using sequences and convergent sequences.
Thus one can ask, “Can I detect compactness using probes from , and ?”. The short answer to this is “No”, but that just reveals that the question was too restrictive. Rather, one should ask “What does compactness look like if all I’m allowed to use are probes from and ?”. The answer to that question is “sequential compactness”.
Thus sequential compactness is what compactness looks like if all one has to test it are sequences.
The following is a list of properties of and pertaining to sequentially compact spaces.
For a metric space, the notions of sequential compactness and compactness coincide.
The Eberlein–Šmulian theorem? states that in a Banach space, for a subset with regard to the weak topology, compactness and sequentially compactness are both equivalent to the weaker notion of countable compactness?.
A countable product of sequentially compact spaces is again sequentially compact.
Let be a countable family of sequentially compact spaces. Let be a sequence in . For each we recursively define an infinite subset with the property that the sequence converges when projected down to . Let . Consider the sequence . For each , we choose a limit of the projection of to . Let . Let be a neighbourhood of . Then there is some and neighbourhood of such that contains the preimage of . For , the sequence is contained in and so the image of converges to . Hence there is some such that for , the projection of lies in . Hence for , . Thus converges to and so is sequentially compact.
This shows that the example of a compact space that is not sequentially compact is about as simple as can be.
The theorem that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism has a counterpart for sequentially compact spaces.
Let and be two topologies on a set such that:
Let be such that . Then it must be non-empty and there must be a point such that is not a neighbourhood of . As is completely regular and singleton sets are sets, there is a continuous function such that . Since is not a neighbourhood of , for each , the set is not wholly contained in . Thus for each there is a point such that and . As is sequentially compact, this sequence has a -convergent subsequence, say converging to . Since , and thus . Thus and so in . As for all , and , it must be the case that is not a -neighbourhood of . Hence . Thus , whence they are equal.
Compactness does not imply sequentially compactness, nor does sequentially compactness imply compactness, without further assumptions (see for example wikipedia: compact spaces). In metric spaces for example both notions coincide.
This is not a contradiction to the statement that compact is equivalent to every net having a convergent subnet: Given a sequence in a compact space, its convergent subnet need not be a subsequence (see net for a definition of subnet).
A famous example of a space that is compact, but not sequentially compact, is the product space
Points of are functions , and the product topology is the topology of pointwise convergence.
Define a sequence in with being the nth digit in the binary expansion of (we make the convention that binary expansions do not end in sequences of s). Let be a subsequence and define by the binary expansion that has a in the th position if is even and a if is odd (and, for definiteness and to ensure that this fits with our convention, define it to be elsewhere). Then the projection of at the th coordinate is which is not convergent. Hence is not convergent.
(Basically that’s the diagonal trick of Cantor's theorem).
However, as is compact, has a convergent subnet. An explicit construction of a convergent subset, given a cluster point , is as follows. A function is a cluster point of if, for any the set
is infinite. We index our subnet by the family of finite subsets of and define the subnet function to be
This is a convergent subnet.
This counterexample is based on the one in item 105 of the book