A counterexample to Wood's conjecture

Abstract

We prove that if the one-point compactification of a locally compact, noncompact Hausdorff space L is the topological space called pseudoarc, then C 0 (L, C) is almost transitive. We also obtain two necessary conditions on a metrizable locally compact Hausdorff space L for C 0 (L) being almost transitive.

ALMOST TRANSITIVITY IN $\mathcal{C}_0$ SPACES OF VECTOR-VALUED FUNCTIONS

“…In [18] and [10] it is proved that if L is a locally compact, non-compact space andL is the topological space known as a pseudoarc, then L is an L 0 -space. However, such an L cannot be an L 1 -space because the pseudoarc is metrizable.…”

Local isometries on spaces of continuous functions

“…By the main result of [19] (Theorem 4.4) this implies that, given f ∈ C R 0 (P a) and ε > 0, there is a homeomorphism ϕ of P leaving a fixed and such that…”

“…Very recently, Fernando Rambla [19] has shown that there are locally compact spaces L for which the complex space C 0 (L) is almost isotropic, thus solving a long-standing problem posed by Wood in [24]. Rambla uses the pseudoarc, a certain continuum constructed by Bing in [2], to solve the problem.…”

“…Rambla shows in [19,Corollary 4.5] that the complex space C 0 (P p) is almost isotropic for all p ∈ P . We are now ready for the counterexamples.…”

Convex-transitivity and function spaces

“…In this article we will investigate the abovementioned transitivity conditions, mainly convex-transitivity, in different settings. For some important results related to this investigation, see [12,13,18,26].…”