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.

Projective Elements in Categories with Perfect θ-Continuous Maps

1981 ◽  
Vol 33 (4) ◽  
pp. 872-884 ◽  
Author(s):  
Hans Vermeer ◽  
Evert Wattel
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
List Type ◽  
Hausdorff Spaces ◽  
Extremally Disconnected ◽  
Continuous Maps

In 1958 Gleason [6] proved the following :THEOREM. In the category of compact Hausdorff spaces and continuous maps, the projective elements are precisely the extremally disconnected spaces.The projective elements in many topological categories with perfect continuous functions as morphisms have been found since that time. For example: In the following categories the projective elements are precisely the extremally disconnected spaces:(i) The category of Tychonov spaces and perfect continuous functions. [4] [11].(ii) The category of regular spaces and perfect continuous functions. [4] [12].(iii) The category of Hausdorff spaces and perfect continuous functions. [10] [1].(iv) In the category of Hausdorff spaces and continuous k-maps the projective members are precisely the extremally disconnected k-spaces. [14].In 1963 Iliadis [7] constructed for every Hausdorff space X the so called Iliadis absolute E[X], which is a maximal pre-image of X under irreducible θ-continuous maps.

scholarly journals Separability of path spaces under the open-point and bi-point-open topologies

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2209-2213
Author(s):  
Anubha Jindal
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
Tychonoff Space ◽  
Continuous Maps ◽  
Open Point ◽  
Made In

In [3], two new kinds of topologies called the open-point topology and the bi-point-open topology on C(X), the set of all real-valued continuous functions on a Tychonoff space X, have been introduced. In this article, we study the separability of the space P(X), of all continuous maps on [0; 1] into a Hausdorff space X, with the open-point and bi-point-open topologies. Our result also demonstrates, the claim made in [3], that both the domain as well as the codomain play significant roles in the construction of the open-point and bi-point-open topologies.

scholarly journals A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 250-263
Author(s):  
V. Mykhaylyuk ◽  
O. Karlova
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Baire Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
Urysohn Space ◽  
Completely Regular

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

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.

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 An Extension of the Concept of the Order Dual of a Riesz Space

1967 ◽  
Vol 19 ◽  
pp. 488-498 ◽  
Author(s):  
W. A. J. Luxemburg ◽  
J. J. Masterson
Keyword(s):  
Hilbert Space ◽  
Topological Space ◽  
Hausdorff Space ◽  
Dense Subset ◽  
Multiplication Operator ◽  
Closed Operator ◽  
Riesz Space ◽  
Open Dense Subset ◽  
Representation Space ◽  
Totally Disconnected

Let L be a σ-Dedekind complete Riesz space. In (8), H. Nakano uses an extension of the multiplication operator on a Riesz space into itself (analagous to the closed operator on a Hilbert space) to obtain a representation space for the Riesz space L. He calls such an operator a “dilatator operator on L.” More specifically, he shows that the set of all dilatator operators , when suitable operations are defined, is a Dedekind complete Riesz space which is isomorphic to the space of all functions defined and continuous on an open dense subset of some fixed totally disconnected Hausdorff space. The embedding of L in the function space is then obtained by showing that L is isomorphic to a Riesz subspace of . Moreover, when L is Dedekind complete, it is an ideal in , and the topological space is extremally disconnected.

On l1 subspaces of Orlicz vector-valued function spaces

1987 ◽  
Vol 101 (1) ◽  
pp. 107-112 ◽  
Author(s):  
Fernando Bombal
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Function Spaces ◽  
Complete Characterization ◽  
Complemented Subspace ◽  
Vector Valued Function ◽  
Vector Valued ◽  
General Banach Space

The purpose of this paper is to characterize the Orlicz vector-valued function spaces containing a copy or a complemented copy of l1. Pisier proved in [13] that if a Banach space E contains no copy of l1, then the space Lp(S, Σ, μ, E) does not contain it either, for 1 < p < ∞. We extend this result to the case of Orlicz vector valued function spaces, by reducing the problem to the situation considered by Pisier. Next, we pass to study the problem of embedding l1 as a complemented subspace of LΦ(E). We obtain a complete characterization when E is a Banach lattice and only partial results in case of a general Banach space. We use here in a crucial way a result of E. Saab and P. Saab concerning the embedding of l1 as a complemented subspace of C(K, E), the Banach space of all the E-valued continuous functions on the compact Hausdorff space K (see [14]). Finally, we use these results to characterize several classes of Banach spaces for which LΦ(E) has some Banach space properties, namely the reciprocal Dunford-Pettis property and Pelczyński's V property.

scholarly journals Weaklyα-continuous functions

1987 ◽  
Vol 10 (3) ◽  
pp. 483-490 ◽  
Author(s):  
Takashi Noiri
Keyword(s):  
Continuous Function ◽  
Hausdorff Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Weak Continuity ◽  
Topological Spaces ◽  
Connected Spaces

In this paper, we introduce the notion of weaklyα-continuous functions in topological spaces. Weakα-continuity and subweak continuity due to Rose [1] are independent of each other and are implied by weak continuity due to Levine [2]. It is shown that weaklyα-continuous surjections preserve connected spaces and that weaklyα-continuous functions into regular spaces are continuous. Corollary1of [3] and Corollary2of [4] are improved as follows: Iff1:X→Yis a semi continuous function into a Hausdorff spaceY,f2:X→Yis either weaklyα-continuous or subweakly continuous, andf1=f2on a dense subset ofX, thenf1=f2onX.

scholarly journals Gelfand theorem implies Stone representation theorem of Boolean rings

1995 ◽  
Vol 18 (4) ◽  
pp. 701-704
Author(s):  
Parfeny P. Saworotnow
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Representation Theorem ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Boolean Rings ◽  
Stone Representation ◽  
Stone Theorem

Stone Theorem about representing a Boolean algebra in terms of open-closed subsets of a topological space is a consequence of the Gelfand Theorem about representing aB∗- algebra as the algebra of continuous functions on a compact Hausdorff space.

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.

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