A Hilbert cube compactification of the function space with the compact-open topology

• Autorzy: Atsushi Kogasaka , Katsuro Sakai
• Języki publikacji: EN

Abstrakty

EN

Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $$ \bar C $$(X) of C(X) such that the pair ($$ \bar C $$(X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification $$ \bar C $$(X) coincides with the space USCCF(X,) of all upper semi-continuous set-valued functions φ: X → = [−∞, ∞] such that each φ(x) is a closed interval, where the topology for USCCF(X, ) is inherited from the Fell hyperspace Cld*F(X × ) of all closed sets in X × .

Słowa kluczowe

EN

The Hilbert cube; The psuedo-interior; The psuedo-boundary; Compactification; Function space; The compact-open topology; The Fell topology

Kategorie tematyczne

• 54C35: Function spaces
• 54C60: Set-valued maps
• 54B20: Hyperspaces
• 57N20: Topology of infinite-dimensional manifolds
• 46E10: Topological linear spaces of continuous, differentiable or analytic functions
• 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2009
• Tom: 7
• Numer: 4
• Strony: 670-682

Daty

wydano

2009-12-01

online

2009-10-31

Twórcy

autor

Atsushi Kogasaka

• Graduate School of Mathematical Sciences, University of Tsukuba, Tsukuba, 305-8571, Japan

autor

Katsuro Sakai

• Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-009-0041-4