The controlled separable complementation property and monolithic compacta

Abstract

For a compact K, a necessary condition for C(K) to have the Controlled Separable Complementation Property is that K be monolithic. In this paper, we prove that when K contains no copy of [0, ω ω ] and the set of points which admit a countable neighborhood base is a cofinite subset of K, then monolithicity of K is sufficient for C(K) to enjoy the Controlled Separable Complementation Property. We also show that, for this type of compacta K, the space C(K) is separably extensible.

On the c_0-extension property

“…Note that Proposition 2.9 provides an example of a space with the SCP but not the c 0 EP. A stronger property than the SCP is the controlled separable complementation property (CSCP) that was introduced in [26] and studied in a series of papers [11,12,13]. A surprising consequence of Theorem 2.5 is that the CSCP implies the c 0 EP.…”

The Separable Complementation Property and Mrówka Compacta

“…& Wójtowicz, M., 2011;Ferrer, J., Koszmider, P. & W. Kubiś, 2013;Ferrer, J., 2014;Ferrer, J., 2009), we studied the controlled version of the separable complementation property (CS CP, for short) for general Banach spaces and in particular for C(K A ) spaces when K A is the Mrówka compact associated to an almost disjoint family A of countable subsets of a given set. After seeing that K being monolithic, see (Arkhangel'skii, A. V., 1992), is a necessary condition in order that the space C(K) enjoys the CS CP, we proved this condition to be sufficient when K is a Mrówka compact and moreover we also showed that this condition suffices in general when K is a scattered compact such that each of its points, except possibly the ones in the top layer, admit a countable neighborhood base.…”

On the $c_0$-extension property

“…Note that Proposition 2.8 provides an example of a space with the SCP but not the c 0 EP. A stronger property than the SCP is the controlled separable complementation property (CSCP) that was introduced in [26] and studied in a series of papers [12,11,13]. A surprising consequence of Theorem 2.5 is that the CSCP implies the c 0 EP.…”