Relations approximated by continuous functions in the Vietoris topology

• Autorzy: L'. Holá , R. A. McCoy
• Języki publikacji: EN

Abstrakty

EN

Let X be a Tikhonov space, C(X) be the space of all continuous real-valued functions defined on X, and CL(X×ℝ) be the hyperspace of all nonempty closed subsets of X×ℝ. We prove the following result: Let X be a locally connected locally compact paracompact space, and let F ∈ CL(X×ℝ). Then F is in the closure of C(X) in CL(X×ℝ) with the Vietoris topology if and only if: (1) for every x ∈ X, F(x) is nonempty; (2) for every x ∈ X, F(x) is connected; (3) for every isolated x ∈ X, F(x) is a singleton set; (4) F is upper semicontinuous; and (5) F forces local semiboundedness. This gives an answer to Problem 5.5 in [HM] and to Question 5.5 in [Mc2] in the realm of locally connected locally compact paracompact spaces.

Kategorie tematyczne

• 54C35: Function spaces
• 54C05: Continuous maps
• 54C60: Set-valued maps
• 54B20: Hyperspaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2007
• Tom: 195
• Numer: 3
• Strony: 205-219

Daty

wydano

2007

Twórcy

autor

L'. Holá

• Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia

autor

R. A. McCoy

• Virginia Polytechnic Institute and State University, Department of Mathematics, Blacksburg, VA 24061, U.S.A.

Identyfikatory

DOI

10.4064/fm195-3-2