Characterization of (semi-)Eberlein compacta using retractional skeletons

Abstract

We deeply study retractions associated to suitable models in compact spaces admitting a retractional skeleton and find several interesting consequences. Most importantly, we provide a new characterization of Valdivia compacta using the notion of retractional skeletons, which seems to be helpful when characterizing its subclasses. This characterization of Valdivia compacta allowed us to characterize Eberlein and semi-Eberlein compacta in terms of retractional skeletons.

Small semi-Eberlein compacta and inverse limits

“…Proof. Item (1) follows from Skolem's Theorem ( [CCS,Theorem 4]) and [CCS,Proposition 17]. Now let us prove (2).…”

“…Note that if a compact space is small, then the characterization of when it is semi-Eberlein presented in [CCS,Corollary 32] is much simpler then the general characterization given by [CCS, Theorem B]. This allows us to obtain in Theorem 2.6 an interesting characterization of small semi-Eberlein spaces in terms of inverse limits.…”

“…The theory developed in [CCS] allows us to show in Lemma 2.3 that every compact space that admits a retractional skeleton is the inverse limit of a continuous inverse system such that the bonding maps and the projections are retractions with nice properties. The key ingredient of Lemma 2.3 is the notion of canonical retractions associated to suitable models that was introduced in [CCS,Definition 12].…”

“…(1) There exists a family of sets (M α ) α∈κ satisfying conditions (Ra)-(Rd) of [CCS,Proposition 17] and for each α ∈ κ, there exists the canonical retraction r α : K → K associated to M α , K and D(s). Moreover, conditions (R1)-(R10) of [CCS,Proposition 17] are satisfied.…”

“…A fundamental concept involved in the proofs of Theorems 2.4 and 2.6 is the shrinkingness of a retractional skeleton that was introduced in [CCS,Definition 29]. Recall that given a retractional skeleton s = (R s ) s∈Σ on a compact space K, a bounded subset A of C(K) and a subset D of K, we say that s is A-shrinking with respect to D if for every x ∈ D and every increasing sequence…”