Generalized Metrizable Spaces

Topology and Its Applications, 2006

We show that the cardinality of a ccc-space with a regular G δ -diagonal is at most 2 ω .

Topology and its Applications, 2002

We obtain several results and examples concerning the general question "When must a space with a small diagonal have a G δ-diagonal?". In particular, we show (1) every compact metrizably fibered space with a small diagonal is metrizable; (2) there are consistent examples of regular Lindelöf (even hereditarily Lindelöf) spaces with a small diagonal but no G δ-diagonal; (3) every first-countable hereditarily Lindelöf space with a small diagonal has a G δ-diagonal; (4) assuming CH, every Lindelöf Σ-space with a small diagonal has a countable network; (5) the statement "countably compact spaces with a small diagonal are metrizable" is consistent with and independent of ZF C; (6) there is in ZF C a locally compact space with a small diagonal but no G δ diagonal. 1991 Mathematics Subject Classification. 54D99. Key words and phrases. small diagonal, G δ-diagonal, countably compact, locally compact, Lindelöf. Research partially supported by NSF DMS-9704849 1 Hušek actually used the more descriptive "ω 1-inaccessible diagonal", but the term "small diagonal", which was suggested by E. van Douwen, seems to have become more popular.

2019

We introduce a new class of -metrizable spaces, namely countably -metrizable spaces. We show that the class of all -metrizable spaces is a proper subclass of counably -metrizable spaces. On the other hand, for pseudocompact spaces the new class coincides with -metrizable spaces. We prove a generalization of a Chigogidze result that the Čech-Stone compactification of a pseudocompact countably -metrizable space is -metrizable.

Topology And Its Applications, 1975

Applied General Topology, 2013

In this paper, a new class of sets called µ-generalized closed (briefly µg-closed) sets in generalized topological spaces are introduced and studied. The class of all µg-closed sets is strictly larger than the class of all µ-closed sets (in the sense ofÁ. Császár). Furthermore, g-closed sets (in the sense of N. Levine) is a special type of µg-closed sets in a topological space. Some of their properties are investigated here. Finally, some characterizations of µ-regular and µ-normal spaces have been given.

Journal of Mathematics

A space X is said to be set selectively star-ccc if for each nonempty subset B of X , for each collection U of open sets in X such that B ¯ ⊂ ∪ U , and for each sequence A n : n ∈ ℕ of maximal cellular open families in X , there is a sequence A n : n ∈ ℕ such that, for each n ∈ ℕ , A n ∈ A n and B ⊂ St ∪ n ∈ ℕ A n , U . In this paper, we introduce set selectively star-ccc spaces and investigate the relationship between set selectively star-ccc and other related spaces. We also study the topological properties of set selectively star-ccc spaces. Some open problems are posed.

Fundamenta Mathematicae, 2001

Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power X ω of any subspace X ⊂ Y is not universal for the class A 2 of absolute G δσ-sets; moreover, if Y is an absolute F σδ-set, then X ω contains no closed topological copy of the Nagata space N = W (I, P); if Y is an absolute G δ-set, then X ω contains no closed copy of the Smirnov space σ = W (I, 0). On the other hand, the countable power X ω of any absolute retract of the first Baire category contains a closed topological copy of each σ-compact space having a strongly countable-dimensional completion. We also prove that for a Polish space X and a subspace Y ⊂ X admitting an embedding into a σ-compact sigma hereditarily disconnected space Z the weak product W (X, Y) = {(x i) ∈ X ω : almost all x i ∈ Y } ⊂ X ω is not universal for the class M 3 of absolute G δσδ-sets; moreover, if the space Z is compact then W (X, Y) is not universal for the class M 2 of absolute F σδ-sets. A topological space X is called C-universal, where C is a class of spaces, if for every space C ∈ C there is a closed embedding f : C → X. It is well known that the Hilbert cube Q = [0, 1] ω is M 0-universal, whereas its pseudointerior s = (0, 1) ω is M 1-universal, where M 0 and M 1 are the Borel classes of compact and Polish spaces, respectively (all spaces considered in this paper are metrizable and separable, all maps are continuous). Let us remark that both Q and s are countable products of finite-dimensional spaces. This raises the following question: can the countable power X ω of a finite-dimensional space X be C-universal for a higher Borel class C? Taking into account results of [BR] and [Ca 1 ], it was conjectured in [Ba] that the

Topology and its Applications, 1980

In this paper a new class of sets called regular generalized δ-closed set (briefly rgδ-closed set)is introduced and its properties are studied. Several examples are provided to illustrate the behaviour of these new class of sets.

2008

We introduce and study (metrically) quarter-stratifiable spaces and then apply them to generalize Rudin and Kuratowski-Montgomery theorems about the Baire and Borel complexity of separately continuous functions. The starting point for writing this paper was the desire to improve the results of V.K. Maslyuchenko et al. [MMMS], [MS], [KM], [KMM] who generalized a classical theorem of W.Rudin [Ru] which states that every separately continuous function f : X × Y → R on the product of a metrizable space X and a topological space Y belongs to the first Baire class. It was proven in [MMMS] that the metrizability of X in the Rudin theorem can be weakened to the σ-metrizability and paracompactness of X. A subtle analysis of Rudin's original proof reveals that this theorem is still valid for a much wider class of spaces X. These spaces are of independent interest, so we decided to give them a special name-metrically quarter-stratifiable spaces. (Metrically) quarter-stratifiable spaces are introduced and studied in details in the first three sections of this paper, where we investigate relationships between the class of (metrically) quarter-stratifiable spaces and other known classes of generalized metric spaces. It turns out that each semi-stratifiable space is quarter-stratifiable (this is a reason for the choice of the term "quarter-stratifiable"), while each quarter-stratifiable Hausdorff space has G δ-diagonal. Because of this, the class of quarter-stratifiable spaces is "orthogonal" to the class of compacta-their intersection contains only metrizable compacta. The class of quarter-stratifiable spaces is quite wide and has many nice inheritance properties. Moreover, every (submetrizable) space with G δ-diagonal is homeomorphic to a closed subset of a (metrically) quarter-stratifiable T 1-space. The following diagram describes the interplay between the class of (metrically) quarter-stratifiable spaces and other classes of generalized metric spaces in the framework of Hausdorff spaces.