Feeble regular and feeble normal spaces in $alpha$-topological spaces using graph

In this paper we introduce (∈ − 2)-ball centered at each point in 2-metric topological space (X,d). Theorems on the normal, regular and Hausdorff topological spaces in 2-metrizable topological space are presented. We show that every metrizable topological spaces are coarser than 2-metrizable topological space, and then we conclude that each manifold is a 2-metric topological space.

A topological space is called a uqu space [10] if it admits a unique quasi-uniformity. Answering a question [2, Problem B, p. 45] of P. Fletcher and W. F. Lindgren in the affirmative we show in [8] that a topological space X is a uqu space if and only if every interior-preserving open collection of X is finite. (Recall that a collection ℒ of open sets of a topological space is called interior-preserving if the intersection of an arbitrary subcollection of ℒ is open (see e.g. [2, p. 29]).) The main step in the proof of this result in [8] shows that a topological space in which each interior-preserving open collection is finite is a transitive space. (A topological space is called transitive (see e.g. [2, p. 130]) if its fine quasi-uniformity has a base consisting of transitive entourages.) In the first section of this note we prove that each hereditarily compact space is transitive. The result of [8] mentioned above is an immediate consequence of this fact, because, obviously, a topological space in which each interior-preserving open collection is finite is hereditarily compact; see e.g. [2, Theorem 2.36]. Our method of proof also shows that a space is transitive if its fine quasi-uniformity is quasi-pseudo-metrizable. We use this result to prove that the fine quasi-uniformity of a T1 space X is quasi-metrizable if and only if X is a quasi-metrizable space containing only finitely many nonisolated points. This result should be compared with Proposition 2.34 of [2], which says that the fine quasi-uniformity of a regular T1 space has a countable base if and only if it is a metrizable space with only finitely many nonisolated points (see e.g. [11] for related results on uniformities). Another by-product of our investigations is the result that each topological space with a countable network is transitive.

e-2 - Classical Metrization Theorems

Soft ω-local indiscreetness as a weaker form of both soft local countability and soft local indiscreetness is introduced. Then soft ω-regularity as a weaker form of both soft regularity and soft ω-local indiscreetness is defined and investigated. Additionally, soft ω-T2 as a new soft topological property that lies strictly between soft T2 and soft T1 is defined and investigated. It is proved that soft anti-local countability is a sufficient condition for equivalence between soft ω-locally indiscreetness (resp. soft ω-regularity) and soft locally indiscreetness (resp. soft ω-regularity). Additionally, it is proved that the induced topological spaces of a soft ω-locally indiscrete (resp. soft ω-regular, soft ω-T2) soft topological space are (resp. ω-regular, ω-T2) topological spaces. Additionally, it is proved that the generated soft topological space of a family of ω-locally indiscrete (resp. ω-regular, ω-T2) topological spaces is soft ω-locally indiscrete and vice versa. In addition to these, soft product theorems regarding soft ω-regular and soft ω-T2 soft topological spaces are obtained. Moreover, it is proved that soft ω-regular and soft ω-T2 are hereditarily under soft subspaces.

We introduce and investigate the concepts of θ ω -limit points and θ ω -interior points, and we use them to introduce two new topological operators. For a subset B of a topological space Y , σ , denote the set of all limit points of B (resp. θ -limit points of B , θ ω -limit points of B , interior points of B , θ -interior points of B , and θ ω -interior points of B ) by D B (resp. D θ B , D θ ω B , Int B , Int θ B , and Int θ ω B ). Several results regarding the two new topological operators are given. In particular, we show that D θ ω B lies strictly between D B and D θ B and Int θ ω B lies strictly between Int θ B and Int B . We show that D B = D θ ω B (resp. Cl θ B = Cl θ ω B and D B = D θ ω B = D θ B ) for locally countable topological spaces (resp. antilocally countable topological spaces and regular topological spaces). In addition to these, we introduce several product theorems concerning metacompactness.

This article continues the study of computable elementary topology started by the author and T. Grubba in 2009 and extends the author's 2010 study of axioms of computable separation. Several computable T3- and Tychonoff separation axioms are introduced and their logical relation is investigated. A number of implications between these axioms are proved and several implications are excluded by counter examples, however, many questions have not yet been answered. Known results on computable metrization of T3-spaces from M. Schr/"oder (1998) and T. Grubba, M. Schr/"oder and the author (2007) are proved under uniform assumptions and with partly simpler proofs, in particular, the theorem that every computably regular computable topological space with non-empty base elements can be embedded into a computable metric space. Most of the computable separation axioms remain true for finite products of spaces.

As a weaker form of ω-paracompactness, the notion of σ-ω-paracompactness is introduced. Furthermore, as a weaker form of σ-ω-paracompactness, the notion of feebly ω-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are feebly ω-paracompact. Furthermore, it is proven hereinthat countably ω-paracompact σ-ω-paracompact topological spaces are ω-paracompact. Furthermore, it is proven hereinthat σ-ω-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, is proven hereinthat ω-paracompactness is inverse invariant under perfect mappings with countable fibers. Furthermore, if A is a locally finite closed covering of a topological space X,τ with each A∈A being ω-paracompact and normal, then X,τ is ω-paracompact and normal, and as a corollary, a sum theorem for ω-paracompact normal topological spaces follows. Moreover, three open questions are raised.

Real-life structures always include indeterminacy.The Mathematical tool which is well-known in dealing with indeterminacy is Neutrosophic.Smarandache proposed the approach of Neutrosophic sets.Neutrosophic sets deal with uncertain data.The notion of Neutrosophic set is generally referred to as the generalization of intuitionistic fuzzy set.In 2022, S. Pious Missier, A. Anusuya and J. Martina Jency introduced and studied the concepts of Neutrosophic generalized regular star closed eu closed *-N gr sets and Neutrosic generalized regular star eu open *-N gr sets in Neutrosophic topological spaces and studied some of their properties and characterizations as well as analyzed the relationships between these newly introduced sets and the already existing neutrosophic sets.In this paper, we introduce the notions of eu compact spaces,

In this paper notions ofm-Lindelof, meta-m-Lindelof, para-m-Lindelof andm-closure preserving property are defined, wherem is any infinite cardinal. The main results are the following: (a) A topological space ism-Lindelof if and only if it is meta-m-Lindelof and it ism-Lindelof in the sense of complete accumulation point. (b) A regular topological space is paracompact if and only if it is para-m-Lindelof and it hasm-closure preserving property for somem.

The principal results of the paper are as follows. A topological space with a σ \sigma -point finite base has a σ \sigma -disjoint base if it is either hereditarily collectionwise normal or hereditarily screenable. From a metrization theorem of Arhangel’skiĭ, it follows that a T 1 {T_1} -space with a σ \sigma -point finite base is metrizable iff it is perfectly normal and collectionwise normal. A topological space with a σ \sigma -point base is quasi-developable in the sense of Bennett. Consequently a theorem of Čoban follows that for a topological space ( X , ℑ ) (X,\Im ) the following are equivalent: (a) ( X , ℑ ) (X,\Im ) is a metacompact normal Moore space, (b) ( X , ℑ ) (X,\Im ) is a perfectly normal T 1 {T_1} -space with a σ \sigma -point finite base.

On Computable Metrization

In [1, p. 41, Theorem 3.10] Arhangel'skiï proved that the perfect image of a completely regular space of countable type is of countable type, and he asked [1, p. 60, problem 4] if a similar result held for regular or Hausdorff spaces. In this paper, it is proved that the perfect image of a space of countable type is of countable type, provided that the image is Hausdorff or regular. An affirmative answer to both of Arhangel'skiï's questions follows immediately from this. Arhangel'skiï made use of the Stone-Čech compactification in the proof of his result, but the proofs below are of a different nature.Let X be a topological space and let K ⊂ X. A collection of open sets is called a base at K provided that for every open set W ⊃ K there exists such that K ⊂ U ⊂ W. Clearly, we may assume that every member of contains K.

Additivity of metrizability and related properties

UDC 515.122 We consider the problem of characterization of topological spaces embedded into countably compact Hausdorff topological spaces. We study the separation axioms for subspaces of Hausdorff countably compact topological spaces and construct an example of a regular separable scattered topological space that cannot be embedded into a Urysohn countably compact topological space.

We introduce and investigate three topological spaces ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> (X,\Lambda_m)$, $(X,\Lambda_{mc}^*)$ and $(X,\Lambda_{g\Lambda_m})$ by using $\Lambda_m$-sets, $(\Lambda, m)$-closed sets and generalized $\Lambda_m$-sets, respectively. Especially, we study properties of weak separation axioms on these topological spaces. The investigation enables us to obtain a unified theory of notions related to $\Lambda$-sets [21], semi-$\Lambda$-sets [5] and pre-$\Lambda$-sets [15] in topological spaces.