Set-open Topology Research Articles

Generalization of the Grothendieck's theorem

In this paper, we obtain a generalization of Grothendieck's theorem for the space of continuous mappings Cλ,μ(X,Y) where Y is a complete uniform space with the uniformity μ endowed with the topology of uniform convergence on the family λ of subsets of X. A new topological game is defined - the Asanov-Velichko game, which makes it possible to single out a class of topological spaces of the Grothendieck type.The developed technique is used to generalize the Grothendieck's theorem for the space of continuous mappings endowed with the set-open topology.

Embedding theorems for function spaces

In this paper, we have proved results similar to Tychonoff's Theorem on embedding a space of functions with the topology of pointwise convergence into the Tychonoff product of topological spaces, but applied to the function space C(X,Y) of all continuous functions from a topological space X into a uniform space Y with the topology of uniform convergence on a family of subsets of X and with the (weak) set-open topology. We also investigated the following question: how the topological embedding of the space C(X,Y) is related to algebraic structures (such as topological groups, topological rings and topological vector spaces) on C(X,Y).

Selection games on continuous functions

In this paper we study the selection principle of closed discrete selection, first researched by Tkachuk in [12] and strengthened by Clontz, Holshouser in [3], in set-open topologies on the space of continuous real-valued functions. Adapting the techniques involving point-picking games on X and Cp(X), the current authors showed similar equivalences in [1] involving the compact subsets of X and Ck(X). By pursuing a bitopological setting, we have touched upon a unifying framework which involves three basic techniques: general game duality via reflections (Clontz), general game equivalence via topological connections, and strengthening of strategies (Pawlikowski and Tkachuk). Moreover, we develop a framework which identifies topological notions to match with generalized versions of the point-open game.

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

Abstract For a Tychonoff space X and a family λ of subsets of X, we denote by C λ(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 C λ(X). Our main results state that C λ(X) is Menger if and only if C λ(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.

Necessary and sufficient conditions for admissible set open topologies

Let X and Y be topological spaces and Cα(X,Y) be the set of all continuous functions from X to Y endowed with the set open topology where α is a nonempty family of subsets of X. We obtain criteria in order that set open topology on C(X,Y) to be admissible. We prove that the topology of Cα(X,Y) is admissible if and only if α is a regular family, when Y is an equiconnected topological space. In the case when X is a topological monoid, we study the relationship between the admissibility of the set open topology on C(X,Y) and the continuity of an action of X on C(X,Y). We prove that the set open topology on C(X,Y) is admissible if and only if Cα(X,Y) with the action s.f=(t↦f(st)) is the cofree X-space over Y.

Separability and Sequential Separability of the Space C(X)

The following results are obtained: (1) a criterion for the separability of the space of continuous functions C(X) with the set-open topology; (2) a criterion for the sequential separability of the space Cp(A|X), where A ⊆ X; (3) an answer to Velichko’s question of whether a set-theoretic condition on a metric space X in a criterion for the sequential separability of Cp(X) is necessary.

Set-open topologies on function spaces

<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>

Comparison of some set open and uniform topologies and some properties of the restriction map

Let $X$ be a Tychonoff space, $Y$ an equiconnected space and $C(X,Y)$ be the set of all continuous functions from $X$ to $Y$. In this paper, we provide a criterion for the coincidence of set open and uniform topologies on $C(X,Y)$ when these topologies are defined by a family $\alpha$ consisting of $Y$-compact subsets of $X$. For a subspace $Z$ of a topological space $X$, we also study the continuity and the openness of the restriction map $\pi_{Z}:C(X,Y)\rightarrow C(Z,Y)$ when both $C(X,Y)$ and $C(Z,Y)$ are endowed with the set open topology.

Variations of selective separability and tightness in function spaces with set-open topologies

We study tightness properties and selective versions of separability in bitopological function spaces endowed with set-open topologies.

The bi-compact-open topology on C(X)

For the set C(X) of real-valued continuous functions on a Tychonoff space X, the compact-open topology on C(X) is a “set-open topology”. This paper studies the separation and countability properties of the space C(X) having the topology given by the join of the compact-open topology and an “open-set topology” called the open-point topology, that was introduced in Jindal et al. (Topol. Appl. 187: 62–74, 2015).

The open-point and bi-point-open topologies on [formula omitted

In the definition of a set-open topology on C(X), the set of all real-valued continuous functions on a Tychonoff space X, we use a certain family of subsets of X and open subsets of R. But instead of using this traditional way to define topologies on C(X), in this paper, we adopt a different approach to define two interesting topologies on C(X). We call them the open-point and the bi-point-open topologies and study the separation and countability properties of these topologies.

Some bounds for the pseudocharacter of the space Cα (X,Y)

Let C? (X,Y) be the set of all continuous functions from X into Y endowed with the set-open topology where ? is a hereditarily closed, compact network on X. We obtain that: i-) ? (f,C? (X,Y)? w?c (X)? sup A?? (?(f(A),Y))? sup A?? (w(f(A)))) ii-) ? (f,C? (X,Y)? w?c (X)? pswe(f(X),Y).

Comparison of some set open topologies on [formula omitted

Let X and Y be two topological spaces and C(X,Y) be the set of all continuous functions from X into Y. A criterion for the coincidence of the set open and uniform topologies on C(X,Y) is given. Also the comparison of the set open topologies on C(X,Y) for various families α is studied.

On the σ-countable compactness of spaces of continuous functions with the set-open topology

For a Tychonoff space X, we obtain a criterion of the σ-countable compactness of the space of continuous functions C(X) with the set-open topology. In particular, for the class of extremally disconnected spaces X, we prove that the space C λ(X) is σ-countably compact if and only if X is a pseudocompact space, the set X(P) of all P-points of the space X is dense in X, and the family λ consists of finite subsets of the set X(P).

Topologizing Homeomorphism Groups

This paper surveys topologies, called admissible group topologies, of the full group of self-homeomorphisms <svg style="vertical-align:-2.3205pt;width:42.299999px;" id="M1" height="15.0875" version="1.1" viewBox="0 0 42.299999 15.0875" width="42.299999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x210B" d="M702 678l12 -14l-18 -14q-124 -96 -232 -310q25 10 62 21l53 18l33 43q101 126 187 195q88 70 135 70q57 0 57 -50q0 -78 -98 -145q-86 -58 -206 -105q-174 -251 -174 -334q0 -15 9.5 -25t24.5 -10q55 0 167 129l21 -13q-120 -146 -202 -146q-39 0 -63 24t-24 65&#xA;q0 101 104 262l-107 -41q-56 -110 -132 -193q-75 -82 -145 -109q-31 -11 -56 -11t-41 16.5t-16 43.5q0 129 283 247q17 29 20 33l22 36q1 2 9 13.5l14 20.5q18 28 59 77q34 39 87 86q-5 -1 -20 -1q-44 0 -124 44q-62 34 -102 34q-48 0 -82 -31q-34 -30 -34 -70q0 -44 22 -69&#xA;t66 -25q37 0 65 20.5t56 65.5l18 -9q-54 -107 -141 -107q-59 0 -93 34.5t-34 91.5q0 58 50 105q49 46 125 46q28 0 58 -11q35 -13 84 -38q59 -31 86 -31q42 0 80 19q16 9 42 28zM958 635q0 22 -24 22q-28 0 -86 -63q-56 -60 -128 -160q238 105 238 201zM307 241&#xA;q-110 -50 -165.5 -98t-55.5 -99q0 -24 24 -29q67 0 197 226z" /></g><g transform="matrix(.017,-0,0,-.017,17.01,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,22.892,12.138)"><path id="x1D44B" d="M775 650l-6 -28q-60 -6 -81.5 -16t-61.5 -54l-175 -191l125 -243q30 -58 48.5 -71t82.5 -19l-5 -28h-275l7 28l35 4q31 4 37 12t-6 34l-108 216q-140 -165 -177 -219q-16 -22 -10.5 -30.5t41.5 -13.5l22 -3l-7 -28h-244l8 28q52 4 75 15.5t67 52.5q48 46 206 231&#xA;l-110 215q-26 51 -44 63t-72 17l6 28h250l-6 -28l-27 -4q-30 -5 -35 -10t3 -27q17 -43 95 -190q70 78 154 185q15 21 10 29.5t-33 12.5l-30 4l5 28h236z" /></g><g transform="matrix(.017,-0,0,-.017,36.355,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg> of a Tychonoff space <svg style="vertical-align:-0.0pt;width:13.5875px;" id="M2" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44B"/></g> </svg>, which yield continuity of both the group operations and at the same time provide continuity of the evaluation function or, in other words, make the evaluation function a group action of <svg style="vertical-align:-2.3205pt;width:42.299999px;" id="M3" height="15.0875" version="1.1" viewBox="0 0 42.299999 15.0875" width="42.299999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x210B"/></g><g transform="matrix(.017,-0,0,-.017,17.01,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,22.892,12.138)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,-0,0,-.017,36.355,12.138)"><use xlink:href="#x29"/></g> </svg> on <svg style="vertical-align:-0.0pt;width:13.5875px;" id="M4" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44B"/></g> </svg>. By means of a compact extension procedure, beyond local compactness and in two essentially different cases of rim-compactness, we show that the complete upper-semilattice <svg style="vertical-align:-3.27605pt;width:52.487499px;" id="M5" height="16.275" version="1.1" viewBox="0 0 52.487499 16.275" width="52.487499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.137)"><path id="x2112" d="M608 643l-12 -19q-56 33 -145 33q-87 0 -154 -42q-68 -42 -68 -115q0 -74 66 -116q67 -42 176 -42q21 0 63 5l29 56q34 66 57 103q43 65 106 113q65 50 122 50q41 0 62 -22t21 -60q0 -67 -87 -144q-88 -76 -217 -110q-46 -101 -119 -186q-72 -83 -136 -108&#xA;q71 -23 103 -23q141 0 265 139l19 -14q-131 -156 -281 -156q-84 0 -164 30q-70 -24 -155 -24q-126 0 -126 61q0 24 30 40.5t78 16.5q42 0 85 -16q18 -6 46 -18q26 -11 42 -16q41 26 99 98q58 70 105 158q-26 -3 -53 -3q-50 0 -96.5 10.5t-88.5 31.5t-67 60t-25 90&#xA;q0 78 73 130q74 53 182 53q98 0 165 -44zM890 588q0 44 -49 44q-55 0 -109 -80q-54 -81 -89 -183q46 12 104 45q64 35 103.5 81t39.5 93zM266 35v1q-73 43 -125 43q-67 0 -67 -25q0 -33 87 -33q65 0 105 14z" /></g> <g transform="matrix(.012,-0,0,-.012,16.137,16.212)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g> <g transform="matrix(.017,-0,0,-.017,27.2,12.137)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,33.082,12.137)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,-0,0,-.017,46.545,12.137)"><use xlink:href="#x29"/></g> </svg> of all admissible group topologies on <svg style="vertical-align:-2.3205pt;width:42.299999px;" id="M6" height="15.0875" version="1.1" viewBox="0 0 42.299999 15.0875" width="42.299999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x210B"/></g><g transform="matrix(.017,-0,0,-.017,17.01,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,22.892,12.138)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,-0,0,-.017,36.355,12.138)"><use xlink:href="#x29"/></g> </svg> admits a least element, that can be described simply as a set-open topology and contemporaneously as a uniform topology. But, then, carrying on another efficient way to produce admissible group topologies in substitution of, or in parallel with, the compact extension procedure, we show that rim-compactness is not a necessary condition for the existence of the least admissible group topology. Finally, we give necessary and sufficient conditions for the topology of uniform convergence on the bounded sets of a local proximity space to be an admissible group topology. Also, we cite that local compactness of <svg style="vertical-align:-0.0pt;width:13.5875px;" id="M7" height="11.175" version="1.1" viewBox="0 0 13.5875 11.175" width="13.5875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D44B"/></g> </svg> is not a necessary condition for the compact-open topology to be an admissible group topology of <svg style="vertical-align:-2.3205pt;width:42.299999px;" id="M8" height="15.0875" version="1.1" viewBox="0 0 42.299999 15.0875" width="42.299999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x210B"/></g><g transform="matrix(.017,-0,0,-.017,17.01,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,22.892,12.138)"><use xlink:href="#x1D44B"/></g><g transform="matrix(.017,-0,0,-.017,36.355,12.138)"><use xlink:href="#x29"/></g> </svg>.

Topological–algebraic properties of function spaces with set-open topologies

For a Tychonoff space X, we denote by C λ ( X ) the space of all real-valued continuous functions on X with set-open topology. In this paper, we study the topological–algebraic properties of C λ ( X ) . Our main results state that (1) C λ ( X ) is a topological vector space (a topological group) iff λ is a family of C-compact sets and C λ ( X ) = C λ ′ ( X ) , where λ ′ consists of all C-compact subsets of every set of λ. In particular, if C λ ( X ) is a topological group, then the set-open topology coincides with the topology of uniform convergence on a family λ; (2) a topological group C λ ( X ) is ω-narrow iff λ is a family of metrizable compact subsets of X.

On the coincidence of set-open and uniform topologies

The ℝ-compactness property is studied, and criteria are established for the coincidence of the set-open topology and the topology of uniform convergence on the spaces of continuous functions.

Some upper bounds for density of function spaces

Let C α ( X , Y ) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X which is closed under finite unions. We proved that the density of the space C α ( X , Y ) is at most iw ( X ) ⋅ d ( Y ) where iw ( X ) denotes the i-weight of the Tychonoff space X, and d ( Y ) denotes the density of the space Y when Y is an equiconnected space with equiconnecting function Ψ, and Y has a base consists of Ψ-convex subsets of Y. We also prove that the equiconnectedness of the space Y cannot be replaced with pathwise connectedness of Y. In fact, it is shown that for each infinite cardinal κ, there is a pathwise connected space Y such that π-weight of Y is κ, but Souslin number of the space C k ( [ 0 , 1 ] , Y ) is 2 κ .

Some properties of set-open topologies

The space of continuous real-valued functions with set-open topology is studied. The coincidence of the set-open topologies for various families λ is considered and conditions for a set-open topology to coincide with the uniform topology are investigated. The separation axioms in the spaces C λ(X) are studied. Criteria for the local compactness and σ-compactness of C λ(X) are given.

Topologizing homeomorphism groups of rim-compact spaces

Let X be a Tychonoff space, H ( X ) the group of all self-homeomorphisms of X with the usual composition and e : ( f , x ) ∈ H ( X ) × X → f ( x ) ∈ X the evaluation function. Topologies on H ( X ) providing continuity of the evaluation function are called admissible. Topologies on H ( X ) compatible with the group operations are called group topologies. Whenever X is locally compact T 2 , there is the minimum among all admissible group topologies on H ( X ) . That can be described simply as a set-open topology, further agreeing with the compact-open topology if X is also locally connected. We show the same result in two essentially different cases of rim-compactness. The former one, where X is rim-compact T 2 and locally connected. The latter one, where X agrees with the rational number space Q equipped with the euclidean topology. In the first case the minimal admissible group topology on H ( X ) is the closed-open topology determined by all closed sets with compact boundaries contained in some component of X. Moreover, whenever X is also separable metric, it is Polish. In the rational case the minimal admissible group topology on H ( Q ) is just the closed-open topology. In both cases the minimal admissible group topology on H ( X ) is closely linked to the Freudenthal compactification of X. The Freudenthal compactification in rim-compactness plays a key role as the one-point compactification does in local compactness. In the rational case we investigate whether the fine or Whitney topology on H ( Q ) induces an admissible group topology on H ( Q ) stronger than the closed-open topology.