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

Atsushi Kogasaka, Katsuro Sakai (2009)

Open Mathematics

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 C ¯ (X) of C(X) such that the pair ( C ¯ (X), C(X)) is homeomorphic to (Q, s). In case...

A homological selection theorem implying a division theorem for Q-manifolds

Taras Banakh, Robert Cauty (2007)

Banach Center Publications

We prove that a space M with Disjoint Disk Property is a Q-manifold if and only if M × X is a Q-manifold for some C-space X. This implies that the product M × I² of a space M with the disk is a Q-manifold if and only if M × X is a Q-manifold for some C-space X. The proof of these theorems exploits the homological characterization of Q-manifolds due to Daverman and Walsh, combined with the existence of G-stable points in C-spaces. To establish the existence of such points we prove (and afterward...

A nice class extracted from C p -theory

Vladimir Vladimirovich Tkachuk (2005)

Commentationes Mathematicae Universitatis Carolinae

We study systematically a class of spaces introduced by Sokolov and call them Sokolov spaces. Their importance can be seen from the fact that every Corson compact space is a Sokolov space. We show that every Sokolov space is collectionwise normal, ω -stable and ω -monolithic. It is also established that any Sokolov compact space X is Fréchet-Urysohn and the space C p ( X ) is Lindelöf. We prove that any Sokolov space with a G δ -diagonal has a countable network and obtain some cardinality restrictions on subsets...

A nice subclass of functionally countable spaces

Vladimir Vladimirovich Tkachuk (2018)

Commentationes Mathematicae Universitatis Carolinae

A space X is functionally countable if f ( X ) is countable for any continuous function f : X → ℝ . We will call a space X exponentially separable if for any countable family ℱ of closed subsets of X , there exists a countable set A ⊂ X such that A ∩ ⋂ 𝒢 ≠ ∅ whenever 𝒢 ⊂ ℱ and ⋂ 𝒢 ≠ ∅ . Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has...

A non-archimedean Dugundji extension theorem

Jerzy Kąkol, Albert Kubzdela, Wiesƚaw Śliwa (2013)

Czechoslovak Mathematical Journal

We prove a non-archimedean Dugundji extension theorem for the spaces C * ( X , 𝕂 ) of continuous bounded functions on an ultranormal space X with values in a non-archimedean non-trivially valued complete field 𝕂 . Assuming that 𝕂 is discretely valued and Y is a closed subspace of X we show that there exists an isometric linear extender T : C * ( Y , 𝕂 ) → C * ( X , 𝕂 ) if X is collectionwise normal or Y is Lindelöf or 𝕂 is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace Y of an ultraregular...

Currently displaying 81 – 100 of 341