Behaviour of the Lindelöf Σ-property in iterated function spaces

✍ Scribed by V.V. Tkachuk

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 93 KB
• Volume: 107
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(99)00112-1

✦ Synopsis

We prove that if C p (X) is a Lindelöf Σ-space, then C p,2n+1 (X) is a Lindelöf Σ-space for every natural n. As a consequence, it is established that υC p C p (X) has the Lindelöf Σ-property. This answers Problem 47 of Arhangel'skiȋ (Recent Progress in General Topology, Elsevier Science, 1992). Another consequence is that only the following distribution of the Lindelöf Σ-property is possible in iterated function spaces: (1) C p,n+1 (X) is a Lindelöf Σ-space for every n ∈ ω; (2) C p,n+1 (X) is a Lindelöf Σ-space only for odd n ∈ ω; (3) C p,n+1 (X) is a Lindelöf Σ-space only for even n ∈ ω; (4) for any n ∈ ω the space C p,n+1 (X) is not a Lindelöf Σ-space.

As an application of the developed technique, we prove that, if X is a Tychonoff space such that ω 1 is a caliber for X and C p (X) is a Lindelöf Σ-space, then X has a countable network. This settles Problem 69 of Arhangel'skiȋ (Recent Progress in General Topology,