Eberlein compactum

An Eberlein compactum is a compactum $X$ that is homeomorphic to a subset of a Banach space with the weak topology [a3].

W.A. Eberlein showed [a1] that such spaces are sequentially compact and Fréchet–Urysohn spaces (cf. Sequentially-compact space; Fréchet space).

One has the following structure theorem for Eberlein compacta: For a compactum $X$ the following are equivalent: i) $X$ is an Eberlein compactum; ii) $X$ is homeomorphic to a subset of $c_0(I)$ in the weak topology (or, equivalently, the pointwise topology), for some set $I$; and iii) $X$ has a family $\mathcal{B} = \cup_{n<\omega} \mathcal{B}_n$ of open $F_\sigma$-sets such that each family $\mathcal{B}_n$ is point-finite and for every $x\ne y$ there is a $B \in \mathcal{B}$ containing exactly one of $x$ and $y$.

Here $c_0(I)$ is the Banach space $$ \left\lbrace{ f \in \mathbf{R}^I : \text{for all}\ \epsilon>0 \ \text{the set}\ \{i\in I: |f(i)|\ge\epsilon\}\ \text{is finite}\, }\right\rbrace \ . $$

The class of Eberlein compact spaces is closed under taking closed subspaces, continuous images and countable products. A recent characterization of Eberlein compact spaces reads as follows [a2]: $X$ is an Eberlein compact space if and only if every subspace of $X^2$ is $\sigma$-metacompact, where $\sigma$-metacompactness means that every open covering has an open refinement which is a union of countably many point-finite families (cf. Paracompact space).

A good survey can be found in [a4].

References

[a1]	W.A. Eberlein, "Weak compactness in Banach spaces" Proc. Nat. Acad. Sci. USA , 33 (1947) pp. 51–53
[a2]	G. Gruenhage, "Games, covering properties and Eberlein compacts" Topology Appl. , 23 (1986) pp. 291–297
[a3]	J. Lindenstrauss, "Weakly compact sets - their topological properties and the Banach spaces they generate" , Symp. infinite-dimensional topology , Ann. Math. Studies , 69 , Princeton Univ. Press (1972) pp. 235–276
[a4]	S. Negrepontis, "Banach spaces and topology" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of set-theoretic topology , North-Holland (1984) pp. 1054–1142