scholarly journals Homotopy Characterization of ANR Function Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Jaka Smrekar
Keyword(s):  
Topological Space ◽  
Homotopy Type ◽  
Metric Spaces ◽  
Function Spaces ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Cw Complex ◽  
Continuous Maps ◽  
Compactly Generated

LetYbe an absolute neighbourhood retract (ANR) for the class of metric spaces and letXbe a topological space. LetYXdenote the space of continuous maps fromXtoYequipped with the compact open topology. We show that ifXis a compactly generated Tychonoff space andYis not discrete, thenYXis an ANR for metric spaces if and only ifXis hemicompact andYXhas the homotopy type of a CW complex.

“Topologically Indexed Function Spaces and Adjoint Functors”

1982 ◽  
Vol 25 (2) ◽  
pp. 169-178
Author(s):  
S. B. Niefield
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Topological Spaces ◽  
Product Topology ◽  
Adjoint Functors ◽  
Point Space ◽  
Continuous Maps ◽  
Necessary And Sufficient

AbstractLet Top denote the category of topological spaces and continuous maps. In this paper we discuss families of function spaces indexed by the elements of a topological space T, and their relationship to the characterization of right adjoints Top/S → Top/T, where S is also a topological space. After reducing the problem to the case where S is a one-point space, we describe a class of right adjoints Top → Top/T, and then show that every right adjoint Top → Top/T is isomorphic to one of this form. We conclude by giving necessary and sufficient conditions for a left adjoint Top/T → Top to be isomorphic to one of the form − XTY, where Y is a space over T, and xT denotes the fiber product with the product topology.

scholarly journals A Characterization of the Existence of a Fundamental Bounded Resolution for the Space Cc(X) in Terms of X

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Juan Carlos Ferrando
Keyword(s):  
Continuous Functions ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Bounded Sets

We characterize in terms of the topology of a Tychonoff space X the existence of a bounded resolution for CcX that swallows the bounded sets, where CcX is the space of real-valued continuous functions on X equipped with the compact-open topology.

scholarly journals Closure properties of function spaces

2003 ◽  
Vol 4 (2) ◽  
pp. 255 ◽  
Author(s):  
Ljubisa D.R. Kocinac
Keyword(s):  
Function Spaces ◽  
Compact Open Topology ◽  
Closure Properties ◽  
Tychonoff Space

<p>In this paper we investigate some closure properties of the space Ck(X) of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology.</p>

scholarly journals The Menger and projective Menger properties of function spaces with the set-open topology

2019 ◽  
Vol 69 (3) ◽  
pp. 699-706 ◽  
Author(s):  
Alexander V. Osipov
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Function Spaces ◽  
List Type ◽  
Finite Sets ◽  
Tychonoff Space ◽  
Menger Space

Abstract For a Tychonoff space X and a family λ of subsets of X, we denote by Cλ(X) the space of all real-valued continuous functions on X with the set-open topology. A Menger space is a topological space in which for every sequence of open covers 𝓤1, 𝓤2, … of the space there are finite sets 𝓕1 ⊂ 𝓤1, 𝓕2 ⊂ 𝓤2, … such that family 𝓕1 ∪ 𝓕2 ∪ … covers the space. In this paper, we study the Menger and projective Menger properties of a Hausdorff space Cλ(X). Our main results state that Cλ(X) is Menger if and only if Cλ(X) is σ-compact; Cp(Y | X) is projective Menger if and only if Cp(Y | X) is σ-pseudocompact where Y is a dense subset of X.

Homotopy Groups and CW Complexes

2020 ◽  
pp. 11-20
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Space ◽  
Algebraic Topology ◽  
Weak Topology ◽  
Homotopy Theory ◽  
Fiber Bundles ◽  
Homotopy Groups ◽  
Cw Complex ◽  
Continuous Maps ◽  
Cw Complexes ◽  
Set Of Points

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.

General function spaces, products and continuous lattices

1986 ◽  
Vol 100 (2) ◽  
pp. 193-205 ◽  
Author(s):  
John Isbell
Keyword(s):  
General Result ◽  
Metric Spaces ◽  
Function Spaces ◽  
Partial Result ◽  
General Function ◽  
Compact Open Topology ◽  
Locally Compact ◽  
Compact Spaces ◽  
Continuous Lattices ◽  
Locally Compact Spaces

The compact–open topology for function spaces is usually attributed to R. H. Fox in 1945 [16]; and indeed, there is no earlier publication to attribute it to. But it is clear from Fox's paper that the idea of the compact–open topology, and its notable success in locally compact spaces, were already familiar. The topology of course goes back to Riemann; and to generalize to locally compact spaces needs only a definition or two. The actual contributions of Fox were (1) to formulate the partial result, and the problem of extending it, clearly and categorically; (2) to show that in separable metric spaces there is no extension beyond locally compact spaces; (3) to anticipate, partially and somewhat awkwardly, the idea of changing the category so as to save the functorial equation. (Scholarly reservations: Fox attributes the question to Hurewicz, and doubtless Hurewicz had a share in (1). As for (2), when Fox's paper was published R. Arens was completing a dissertation which gave a more general result [1] – though worse formulated.)

scholarly journals Characterization of pseudocompactness by the topology of uniform convergence on function spaces

1978 ◽  
Vol 26 (2) ◽  
pp. 251-256 ◽  
Author(s):  
R. A. McCoy
Keyword(s):  
Uniform Convergence ◽  
Continuous Functions ◽  
Function Spaces ◽  
Metrizable Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Of Continuous Functions ◽  
Tychonoff Space ◽  
Necessary And Sufficient

AbstractIt is shown that a Tychonoff space X is pseudocompact if and only if for every metrizable space Y, all uniformities on Y induce the same topology on the space of continuous functions from X into Y. Also for certain pairs of spaces X and Y, a necessary and sufficient condition is established in order that all uniformities on Y induce the same topology on the space of continuous functions from X into Y.

scholarly journals Selection principles and games in bitopological function spaces

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4535-4540
Author(s):  
Daniil Lyakhovets ◽  
Alexander Osipov
Keyword(s):  
Pointwise Convergence ◽  
Continuous Functions ◽  
Function Spaces ◽  
Compact Open Topology ◽  
Bitopological Space ◽  
Tychonoff Space ◽  
Selection Principles ◽  
Topology Of Pointwise Convergence ◽  
Selective Separability

For a Tychonoff space X, we denote by (C(X), ?k ?p) the bitopological space of all real-valued continuous functions on X, where ?k is the compact-open topology and ?p is the topology of pointwise convergence. In the papers [6, 7, 13] variations of selective separability and tightness in (C(X),?k,?p) were investigated. In this paper we continue to study the selective properties and the corresponding topological games in the space (C(X),?k,?p).

scholarly journals Selection principles in function spaces with the compact-open topology

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5403-5413 ◽  
Author(s):  
Alexander Osipov
Keyword(s):  
Convergent Sequence ◽  
Continuous Functions ◽  
Function Spaces ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Selection Principles

For a Tychonoff space X, we denote by Ck(X) the space of all real-valued continuous functions on X with the compact-open topology. A subset A ? X is said to be sequentially dense in X if every point of X is the limit of a convergent sequence in A. In this paper, the following properties for Ck(X) are considered. S1(S,S)=> Sfin(S,S) => Sfin(S,D) <=S1(S,D) S1(D,S) => Sfin(D,S) => Sfin(D,D) <= S1(D,D) For example, a space Ck(X) satisfies S1(S,D) (resp., Sfin(S,D)) if whenever (Sn : n ? N) is a sequence of sequentially dense subsets of Ck(X), one can take points fn ? Sn (resp., finite Fn ? Sn) such that {fn : n ? N} (resp.,U {Fn : n ? Ng) is dense in Ck(X). Other properties are defined similarly. In [22], we obtained characterizations these selection properties for Cp(X). In this paper, we give characterizations for Ck(X).

scholarly journals Some Function Spaces Relative to Morrey-Campanato Spaces on Metric Spaces

2005 ◽  
Vol 177 ◽  
pp. 1-29 ◽  
Author(s):  
Dachun Yang
Keyword(s):  
Sobolev Spaces ◽  
Maximal Function ◽  
Metric Spaces ◽  
Function Spaces ◽  
Sharp Maximal Function ◽  
Embedding Theorems ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Lizorkin Space

In this paper, the author introduces the Morrey-Campanato spaces Lsp(X) and the spaces Cps(X) on spaces of homogeneous type including metric spaces and some fractals, and establishes some embedding theorems between these spaces under some restrictions and the Besov spaces and the Triebel-Lizorkin spaces. In particular, the author proves that Lsp(X) = Bs∞,∞(X) if 0 < s < ∞ and µ(X) < ∞. The author also introduces some new function spaces Asp(X) and Bsp(X) and proves that these new spaces when 0 < s < 1 and 1 < p < ∞ are just the Triebel-Lizorkin space Fsp,∞(X) if X is a metric space, and the spaces A1p(X) and B1p(X) when 1 < p < ∞ are just the Hajłasz-Sobolev spaces W1p(X). Finally, as an application, the author gives a new characterization of the Hajłasz-Sobolev spaces by making use of the sharp maximal function.

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