Generalized Metrizable Spaces

Gruenhage, Gary

doi:10.2991/978-94-6239-024-9_10

Cited by 16 publications

(8 citation statements)

References 111 publications

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“…Degrees of points: submetrizable topology. We say that a space is submetrizable if it admits a continuous metric, that is, either it is metrizable or it has a coarser metrizable topology (see [23]). Every submetrizable space is completely Hausdorff, and every completely Hausdorff space is T 2.5 : submetrizable =⇒ completely Hausdorff =⇒ T 2.5 .…”

Section: 4mentioning

confidence: 99%

Enumeration degrees and non-metrizable topology

Kihara,

Ng,

Pauly

2019

Preprint

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The enumeration degrees of sets of natural numbers can be identified with the degrees of difficulty of enumerating neighborhood bases of points in a universal second-countable T 0 -space (e.g. the ω-power of the Sierpiński space). Hence, every represented second-countable T 0 -space determines a collection of enumeration degrees. For instance, Cantor space captures the total degrees, and the Hilbert cube captures the continuous degrees by definition. Based on these observations, we utilize general topology (particularly non-metrizable topology) to establish a classification theory of enumeration degrees of sets of natural numbers. ContentsTAKAYUKI KIHARA, KENG MENG NG, AND ARNO PAULY 5.3. Degrees of points: T 2.5 -topology 45 5.4. Degrees of points: submetrizable topology 50 5.5. Degrees of points: G δ -topology 56 5.6. Quasi-Polish topology 59 6. Cs-networks and non-second-countability 62 6.1. Regular-like networks and closure representation 63 6.2. Near quasi-minimality 66 6.3. Borel extension topology 67 7. Proofs for Section 4 69 7.1. T 0 -degrees which are not T 1 69 7.2. T 1 -degrees which are not T 2 84 7.3. T 2 -degrees which are not T 2.5 93 7.4. T 2.5 -degrees which are not T 3 97 8. Open Questions 99 References 101Recall that a subset of ω is d-c.e. if it is the difference of two c.e. sets, and co-d-c.e. if it is the complement of a d-c.e. set, that is, the union A ∪ P of a c.e. set P and co-c.e. set A such that A and P are disjoint. Note that an enumeration degree contains a co-d-c.e. set if and only if it contains a 3-c.e. set. Definition 3.5. Let a be an enumeration degree.(1) We say that a is co-d-CEA if a contains a set of the form (X ⊕ X c ) ⊕ (A ∪ P ) for some X, A, P ⊆ ω such that P and A c are X-c.e., and A and P are disjoint. (2) Generally, we say that a is Γ-above if a contains a set of the form (X ⊕ X c ) ⊕ Z such that Z is Γ in X.(3) We say that a is doubled co-d-CEA if a contains a set of the form (X ⊕ X c ) ⊕ (A ∪ P ) ⊕ (B ∪ N) for some X, A, B, P, N ⊆ ω such that P , N, and A ∪ B c are X-c.e., and that A, B, P and N are pairwise disjoint.

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Section: 4mentioning

confidence: 99%

Enumeration degrees and non-metrizable topology

Kihara,

Ng,

Pauly

2019

Preprint

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“…This characterization shows that the class of k * -metrizable spaces is sufficiently wide and contains all Lašnev spaces (closed images of metrizable spaces), all ℵ 0 -spaces and all ℵ-spaces. k * -metrizable spaces form a new class of generalized metric spaces and have various applications in topological algebra, functional analysis, and measure theory, see [3,9]. From [3], we list some properties of k * -metrizable spaces.…”

Section: Notation and Terminologiesmentioning

confidence: 99%

Countable tightness and k-property of free topological vector spaces

Lin

Liu

2019

Topology and its Applications

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“…More information on generalized metric spaces can be found in the surveys of Gruenhage [10], [11], [12] (for k * -metrizable spaces, see [2]). By Theorem 6.4 in [2], a regular space Y is k * -metrizable if and only if Y is the image of a metrizable space M under a continuous map f : M → X that admits a section s : X → M which preserves precompact sets in the sense that for every compact subset K ⊂ X the set s(K) has compact closure in M .…”

Section: Cc-maslyuchenkomentioning

confidence: 99%

Quasicontinuous and separately continuous functions with values in Maslyuchenko spaces

Банах

2017

Topology and its Applications

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We generalize some classical results about quasicontinuous and separately continuous functions with values in metrizable spaces to functions with values in certain generalized metric spaces, called Maslyuchenko spaces. We establish stability properties of the classes of Maslyuchenko spaces and study the relation of these classes to known classes of generalized metric spaces (such as Piotrowski or Stegall spaces). One of our results says that for any ℵ 0 -space Z and any separately continuous function f : X × Y → Z defined on the product of a topological space X and a second-countable space Y , the set D(f ) of discontinuity points of f has meager projection on X.1991 Mathematics Subject Classification. 54C08, 54E18, 54E20, 54E35, 54E52.

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Generalized Metrizable Spaces

Cited by 16 publications

References 111 publications

Enumeration degrees and non-metrizable topology

Enumeration degrees and non-metrizable topology

Countable tightness and k-property of free topological vector spaces

Quasicontinuous and separately continuous functions with values in Maslyuchenko spaces

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