×

A remark on spaces of bounded continuous functions. (English) Zbl 0685.54010

Let X be a topological space and let \(C^*_ p(X)\) denote the vector space of all bounded continuous real-valued functions on X, endowed with the topology of pointwise convergence. It follows from J. van Mill [Compos. Math. 63, 159-188 (1987; Zbl 0634.54011)] that for countable, non-locally compact metric spaces X and Y, the spaces \(C^*_ p(X)\) and \(C^*_ p(Y)\) are homeomorphic. van Mill also raised the question whether a linear homeomorphism exists. The author shows here that, in general, no linear homeomorphism need exist between \(C^*_ p(X)\) and \(C^*_ p(Y)\). The counterexample uses the spaces N of natural numbers, and T, the ‘Tychonoff plank’.
Reviewer: I.E.Blum

MSC:

54C35 Function spaces in general topology
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Citations:

Zbl 0634.54011
PDFBibTeX XMLCite