scholarly journals Strict topoligies in non-Archimedean function spaces

1984 ◽  
Vol 7 (1) ◽  
pp. 23-33 ◽  
Author(s):  
A. K. Katsaras
Keyword(s):  
Topological Space ◽  
Convex Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Dual Spaces ◽  
Valued Field ◽  
Bounded Continuous Functions

LetFbe a non-trivial complete non-Archimedean valued field. Some locallyF-convex topologies, on the spaceCb(X,E)of all bounded continuous functions from a zero-dimensional topological spaceXto a non-Archimedean locallyF-convex spaceE, are studied. The corresponding dual spaces are also investigated.

Strict Topology on Spaces of Continuous Vector-Valued Functions

1979 ◽  
Vol 31 (4) ◽  
pp. 890-896 ◽  
Author(s):  
Seki A. Choo
Keyword(s):  
Tensor Product ◽  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Locally Convex Space ◽  
Completely Regular ◽  
Vector Valued Functions ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Locally Convex

In this paper, X denotes a completely regular Hausdorff space, Cb(X) all real-valued bounded continuous functions on X, E a Hausforff locally convex space over reals R, Cb(X, E) all bounded continuous functions from X into E, Cb(X) ⴲ E the tensor product of Cb(X) and E. For locally convex spaces E and F, E ⴲ, F denotes the tensor product with the topology of uniform convergence on sets of the form S X T where S and T are equicontinuous subsets of E′, F′ the topological duals of E, F respectively ([11], p. 96). For a locally convex space G , G ′ will denote its topological dual.

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.

scholarly journals The regular open-open topology for function spaces

1996 ◽  
Vol 19 (2) ◽  
pp. 299-302 ◽  
Author(s):  
Kathryn F. Porter
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Regular Space ◽  
Spaces Of Continuous Functions ◽  
Regular Open

The regular open-open topology,Troo, is introduced, its properties for spaces of continuous functions are discussed, andTroois compared toToo, the open-open topology. It is then shown thatTrooonH(X), the collection of all self-homeomorphisms on a topological space,(X,T), is equivalent to the topology induced onH(X)by a specific quasi-uniformity onX, whenXis a semi-regular space.

scholarly journals Duality properties of spaces of non-Archimedean valued functions

1987 ◽  
Vol 42 (1) ◽  
pp. 48-56
Author(s):  
W. Govaerts
Keyword(s):  
Uniform Convergence ◽  
Compact Hausdorff Space ◽  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Complete Characterization ◽  
Bornological Space ◽  
Valued Field ◽  
Better Than

AbstractLet C(X, F) be the space of all continuous functions from the ultraregular compact Hausdorff space X into the separated locally K-convex space F; K is a complete, but not necessarily spherically complete, non-Archimedean valued field and C(X, F) is provided with the topology of uniform convergence on X We prove that C(X, F) is K-barrelled (respectively K-quasibarrelled) if and only if F is K-barrelled (respectively K-quasibarrelled) This is not true in the case of R or C-valued functions. No complete characterization of the K-bornological space C(X, F) is obtained, but our results are, nevertheless, slightly better than the Archimedean ones. Finally, we introduce a notion of K-ultrabornological spaces for K non-spherically complete and use it to study K-ultrabornological spaces C(X, F).

scholarly journals Quasi-uniform convergence topologies on function spaces- Revisited

2017 ◽  
Vol 18 (2) ◽  
pp. 301
Author(s):  
Wafa Khalaf Alqurash ◽  
Liaqat Ali Khan
Keyword(s):  
Topological Space ◽  
Uniform Convergence ◽  
Uniform Space ◽  
Continuous Functions ◽  
Function Spaces

Let X and Y be topological space and F(X,Y) the set of all functions from X into Y. We study various quasi-uniform convergence topologies U_{A} (A⊆P(X)) on F(X,Y) and their comparison in the setting of Y a quasi-uniform space. Further, we study U_{A}-closedness and right K-completeness properties of certain subspaces of generalized continuous functions in F(X,Y) in the case of Y a locally symmetric quasi-uniform space or a locally uniform space.

scholarly journals Set-open topologies on function spaces

2018 ◽  
Vol 19 (1) ◽  
pp. 55
Author(s):  
Wafa Khalaf Alqurashi ◽  
Liaqat Ali Khan ◽  
Alexander V. Osipov
Keyword(s):  
Topological Space ◽  
Uniform Convergence ◽  
Uniform Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Topological Spaces ◽  
General Topological Space

<p>Let X and Y be topological spaces, F(X,Y) the set of all functions from X into Y and C(X,Y) the set of all continuous functions in F(X,Y). We study various set-open topologies tλ (λ ⊆ P(X)) on F(X,Y) and consider their existence, comparison and coincidence in the setting of Y a general topological space as well as for Y = R. Further, we consider the parallel notion of quasi-uniform convergence topologies Uλ (λ ⊆ P(X)) on F(X,Y) to discuss Uλ-closedness and right Uλ-K-completeness properties of a certain subspace of F(X,Y) in the case of Y a locally symmetric quasi-uniform space. We include some counter-examples to justify our comments.</p>

scholarly journals On an inequality of Kolmogorov and Stein

2000 ◽  
Vol 61 (1) ◽  
pp. 153-159 ◽  
Author(s):  
Ha Huy Bang ◽  
Hoang Mai Le
Keyword(s):  
Orlicz Spaces ◽  
Continuous Functions ◽  
Function Spaces ◽  
General Function ◽  
Bounded Continuous Functions

A.N. Kolmogorov showed that, if f, f′, …, f (n) are bounded continuous functions on ℝ, then when 0 < k < n. This result was extended by E.M. Stein to Lebesgue Lp-spaces and by H.H. Bang to Orlicz spaces. In this paper, the inequality is extended to more general function spaces.

Real Functions, Covers and Bornologies

2021 ◽  
Vol 78 (1) ◽  
pp. 199-214
Author(s):  
Lev Bukovský
Keyword(s):  
Topological Space ◽  
Pointwise Convergence ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Upper Semicontinuous Functions ◽  
Covering Properties ◽  
Real Functions ◽  
Topology Of Pointwise Convergence ◽  
Semicontinuous Functions

Abstract The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper semicontinuous functions on X with the topology of pointwise convergence. Dealing with properties of continuous functions C(X), we need shrinkable covers. The results are extended for A-measurable and upper A-semimeasurable functions where A is a family of subsets of X. Similar results for covers respecting a bornology and spaces USC(X) or C(X) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.

Order continuous measures

1964 ◽  
Vol 60 (2) ◽  
pp. 205-207 ◽  
Author(s):  
G. T. Roberts
Keyword(s):  
Topological Space ◽  
Linear Form ◽  
Vector Lattice ◽  
Measure Zero ◽  
Continuous Functions ◽  
Order Convergence ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Continuous Measures ◽  
Order Continuous

1. Objective. It is possible to define order convergence on the vector lattice of all continuous functions of compact support on a locally compact topological space. Every measure is a linear form on this vector lattice. The object of this paper is to prove that a measure is such that every set of the first category of Baire has measure zero if and only if the measure is a linear form which is continuous in the order convergence.

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