On function spaces related to d-spaces

Liu, Bei; Li, Qingguo; Ho, Weng Kin

doi:10.1016/j.topol.2021.107757

Cited by 6 publications

(5 citation statements)

References 6 publications

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“…Given topological spaces X and Y , let T OP (X, Y ) be the set of all continuous functions from X to Y . The Isbell topology on the set T OP (X, Y ) is generated by the subsets of the form Lemma 3.16 [7] For each y ∈ Y ,consider the mapping ξ : Y → [X, Y ] by ξ(y) = ξ y , where ξ y (x) = y for all x ∈ X. Then ξ is continuous.…”

Section: Resultsmentioning

confidence: 99%

The $d^{*}$-space

Chu¹,

Li²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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In this paper, we introduce the concept of $d^{\ast}$-spaces. We find that strong $d$-spaces are $d^{\ast}$-spaces, but the converse does not hold. We give a characterization for a topological space to be a $d^{\ast}$-space. We prove that the retract of a $d^{\ast}$-space is a $d^{\ast}$-space. We obtain the result that for any $T_{0}$ space $X$ and $Y$, if the function space $TOP(X,Y)$ endowed with the Isbell topology is a $d^{\ast}$-space, then $Y$ is a $d^{\ast}$-space. We also show that for any $T_{0}$ space $X$, if the Smyth power space $Q_{v}(X)$ is a $d^{\ast}$-space, then $X$ is a $d^{\ast}$-space. Meanwhile, we give a counterexample to illustrate that conversely, for a $d^{\ast}$-space $X$, the Smyth power space $Q_{v}(X)$ may not be a $d^{\ast}$-space.

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Section: Resultsmentioning

confidence: 99%

The $d^{*}$-space

Chu¹,

Li²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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“…In the following, for topological spaces X, Y and y ∈ Y, c y denotes the constant function from X to Y with value y, i.e., c y (x) = y for all x ∈ X. Lemma 18. (Liu et al 2021, Lemma 3.2) Let X and Y be T 0 spaces. Then, the function…”

Section: Some Basic Lemmasmentioning

confidence: 99%

“…As a special kind of mathematical structures, domains serve as mathematical universes within which people can interpret higher-order functional programming languages, and cartesian closed categories of domains (more generally, certain topological spaces) are appropriate for models of various typed and untyped lambda-calculi and functional programming languages (see Gierz et al 2003). Since whether certain properties of topological spaces are preserved when passing to function spaces is connected with the cartesian closed category of topological spaces, this question has attracted considerable attention in domain theory and non-Hausdorff topology, especially for domains (which are a special kind of topological spaces when endowed with the Scott topology), sober spaces, d-spaces and well-filtered spaces (see Ershov et al 2020;Gierz et al 2003;Liu et al 2021;Xu 2021).…”

Section: Introductionmentioning

confidence: 99%

“…Gierz et al 2003, Lemma II-4.3). Conversely, in Liu et al (2021), the authors showed that if the function space C(X, Y) equipped with the Isbell topology is a d-space, then Y is a d-space. For the well-filteredness, it was proved in Liu et al (2021) that for any core-compact space X and well-filtered space Y, the function space C(X, Y) equipped with the Isbell topology is well-filtered.…”

Section: Introductionmentioning

confidence: 99%

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On function spaces equipped with Isbell topology and Scott topology

Bao²,

Zhang³

2022

Math. Struct. Comp. Sci.

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In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces X and Y, it is proved that the following three conditions are equivalent: (1) the Scott space $\Sigma \mathcal O(X)$ of the lattice of all open sets of X is H-sober; (2) for every H-sober space Y, the function space $\mathbb{C}(X, Y)$ of all continuous mappings from X to Y equipped with the Isbell topology is H-sober; (3) for every H-sober space Y, the Isbell topology on $\mathbb{C}(X, Y)$ has property S with respect to H. One immediate corollary is that for a $T_{0}$ space X, Y is a d-space (resp., well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the Isbell topology is a d-space (resp., well-filtered space). It is shown that for any $T_0$ space X for which the Scott space $\Sigma \mathcal O(X)$ is non-sober, the function space $\mathbb{C}(X, \Sigma 2)$ equipped with the Isbell topology is not sober. The function spaces $\mathbb{C}(X, Y)$ equipped with the Scott topology, the compact-open topology and the pointwise convergence topology are also discussed. Our study also leads to a number of questions, whose answers will deepen our understanding of the function spaces related to H-sober spaces.

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“…([10, Lemma 3.2]) Let X and Y be T 0 spaces. Consider the functionψ I : Y −→ [X → Y ] I , y → c y .Then ψ I is continuous.…”

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confidence: 99%

On Function Spaces Related to H-sober Spaces

Meng¹,

Zhang²,

Xu³

2022

Preprint

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In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and T 0 spaces X and Y , it is proved that Y is H-sober iff the function space C(X, Y ) of all continuous functions f : X −→ Y equipped with the topology of pointwise convergence is H-sober iff the function space C(X, Y ) equipped with the Isbell topology is H-sober. One immediate corollary is that for a T 0 space X, Y is a sober space (resp., d-space, well-filtered space) iff the function space C(X, Y ) equipped with the topology of pointwise convergence is a sober space (resp., d-space, well-filtered space) iff the function space C(X, Y ) equipped with the the Isbell topology is a sober space (resp., d-space, well-filtered space). It is shown that T 0 spaces X and Y , if the function space C(X, Y ) equipped with the compact-open topology is H-sober, then Y is H-sober. The function space C(X, Y ) equipped with the Scott topology is also discussed.

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On function spaces related to d-spaces

Cited by 6 publications

References 6 publications

The $d^{*}$-space

The $d^{*}$-space

On function spaces equipped with Isbell topology and Scott topology

On Function Spaces Related to H-sober Spaces

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