On the weak and pointwise topologies in function spaces

Abstract

For a compact space K we denote by C w (K ) (C p (K )) the space of continuous real-valued functions on K endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose that K is an infinite (metrizable) compact space. Can C w (K ) and C p (K ) be homeomorphic? We show that the answer is "no", provided K is an infinite compact metrizable C-space. In particular our proof works for any infinite compact metrizable finite-dimensional space K .

On the weak and pointwise topologies in function spaces II

“…In this paper we continue the research initiated in [Kr2].For a compact space K, the set of all real-valued continuous functions on K equipped with the supremum norm is a Banach space which we denote by C(K). One can consider two, other than norm, topologies on the set of continuous functions on K: the weak topology i.e.…”

“…It was proved in [Kr2] that if K is an infinite metrizable finite-dimensional compactum, then C p (K) and C w (K) are not homeomorphic. In this section we will extend this result to a certain class of non-metrizable compacta (cf.…”

“…It is not clear however whether these two topologies are always non-homeomorphic. The following problem was addressed in [Kr2]. Problem 1.2.…”

“…It turns out that if K is scattered, then the spaces C p (K)and C w (K) can be topologically distinguished by the Fréchet-Urysohn property or the property (B) which we introduce in Section 5. It is curious that in the case when K is scattered we can distinguish C p (K) and C w (K) by a specific (and quite simple) topological property and we could not do it, for instance, for the unit interval (see [Kr2,Problem 1]), though for scattered K the weak and the pointwise topologies seem to be closest to each other; they coincide on bounded subsets.…”

On complemented copies of the space $c_0$ in spaces $C_p(X,E)$

“…Remark 3.10. Krupski and Marciszewski [26,Corollary 3.2] proved that for all infinite compact spaces X and Y the locally convex spaces C p (X) and C(Y…”