Decomposition, Mapping, and Sum Theorems of ω-Paracompact Topological Spaces

A n-dimensional topological manifold is defined as an n-dimensional local Euclidean space M that is also a second countable space and a Hausdorff space. If a topological space has a countable base, it is referred to as a second countable space. At the same time, continuous function and metric space can help people to understand topological space better. This article will express some basic ideas about continuous functions, metric space, topological space and its properties, and topological manifolds. Moreover, the paper shows some basic ideas of how topological manifolds, topological space and function could be recognized and proved. Through analysis, this paper demonstrates the connection between them, such as using properties of continuous function to prove the definition and properties of topological manifolds and spaces.

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.

In this paper, we introduce sufficient conditions for the non—empty intersection of two set—valued mappings in topological spaces. As applications, some topological minimax inequalities for two functions in which one of them is separately lower (or upper) semicontinuous are given. Finally, by employing our topological intersection theorems for two set—valued mappings, some other minimax inequalities have been derived without separately lower (or upper) sernicontinuity under but with another condition. These results are topological versions of corresponding minimax inequalities for two functions due to Fan (1964) and Sion (1958) in topological vector spaces

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.

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.

In this paper we formulate a new structure of topological spaces which we call it lower normal spaces. This concept of spaces is directly and nature connection with the lower transversal continuous mappings on topological spaces. In this sense, we shall study spaces in which it is possible in the same way to separate two disjoint closed sets by a lower continuous real valued function. Applications in nonlinear functional analysis are considered. The concept of lower normal spaces is closely connected with the concept of normal topological spaces and the results of Alexandroff, Urysohn, Tietze, Lebesgue, Dieudonné, Tychonoff, Lefschetz, and Vietoris.

In this paper, we introduce and study the relationship between two different notions of chaotic maps, namely topological α–chaotic maps, topological θ-chaotic maps and investigate some of their properties in two topological spaces (X, τα) and (X, τθ), τα denotes the α–topology(resp. τθ denotes the θ–topology) of a given topological space (X, τ). The two notions are defined by using the concepts of α-transitive map and θ-transitive map respectively Also, we define and study the relationship between two types of minimal mappings, namely, α - minimal mapping and θ-minimal mapping, The main results are the following propositions: 1). Every topologically α-chaotic map is a chaotic map which implies topologically θ- chaotic map, but the converse not necessarily true. 2). Every α-minimal map is a minimal map which implies θ- minimal map in topological spaces, but the converse not necessarily true.

In reminiscence of Ptak's open mapping theorem, a topological space satisfying the open mapping theorem is called a B r space.This paper is devoted to the study of sums and products of B r spaces in the category of topological spaces.We prove that, in general, sums and products of even two B r spaces need no longer be B r .On the other hand, for any B r space E the sum E θ E is again a B r space.Moreover, since Cech complete spaces are known to be B r , we ask whether a sum E θ F is B r provided that E is Cech complete and F is B r .It turns out that, at least in the framework of complete regularity, the answer to this question is in the positive if and only if F is a Baire space.Introduction.The notion of B r spaces has first been introduced by T. Husain in the categories of locally convex vector spaces (see [HuJ) and topological groups (see [Hu 2 ]).Originally, it goes back to V. Ptak's open mapping theorem (see [Kό], p. 35 ff).A Hausdorff topological space E is called a B r space if every continuous, nearly open bijection / from E onto any Hausdorff space F is in fact open.For a survey of the classical theory of B r spaces we refer the reader to Kόthe's book [Kό], where the locally convex case is treated.The linear topological case is investigated in the lecture notes [AEK].B r groups are considered by several authors.See for instance [Hu 2 ], [Ba], [Gr], [Su].In a purely topological context, B r spaces have first been investigated by Weston in [We], although the term B r space is not used there.Translated into the B r terminology, Weston proved that every completely metrizable topological space is a B r space.His result has been generalized by Byczkowski and Pol in [BP], who proved that every Cech complete topological space is a B r space.In [No] we have further generalized this result proving that every Hausdorff, semi-regular topological space densely containing some Cech complete sub space is in fact a B r space.We just mention another generalization of Byczkowski and Pols' result into a somewhat different direction by Wilhelm (cf.[Wi]).In the present paper we examine the invariance of the class of B r spaces under topological sums and products.It turns out that, in general, the sum of even two B r spaces need not be a B r space.A counterexample is given in §2.In §1 we obtain a positive result stating that the sum E θ E

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.

This paper explores the decomposition of pre--irresolute (pre--irr) maps and their connection to generalized closed sets (g-closed) in topological space (top. sp.). We provide the necessary background on top. sp., continuity, closed, and related concepts. We introduce the definitions of pre--irr maps and g-closed and discuss their basic properties. The decomposition theorem is presented, stating that every pre--irr map can be decomposed into a composition of a pre--continuous (pre- -cont.) map and a b-irr map. Examples and applications are provided to illustrate the behavior of pre--irr maps and the practical implications of the decomposition theorem. We examine the relationships between pre--cont. maps and pre--irr maps, exploring the conditions under which a pre--irr map is also -irr, and provide a proof for this relationship. The properties of pre--cont. maps, - maps, g- pre--cont. maps, and g--cont. maps are discussed, along with proofs and examples to illustrate their applicability. This paper contributes to the understanding of pre--irr maps, their decomposition, and their relationships with g-closed, providing a foundation for further research in top. sp. and map decompositions. Keywords: pre-β-irresolute maps, g-closed, pre-β-cont. maps, b-cont. maps

We present new Furi‐Pera theorems for acyclic maps between topological spaces.

In 1920s R. L. Moore introduced upper semicontinuous and lower semicontinuous decompositions in studying decomposition spaces. Upper semicontinuous decompositions were studied very well by himself and later by R.H. Bing in 1950s. In this paper we consider lower semicontinuous decompositions $\mathcal D$ of a topological space $X$ such that the decomposition spaces $X/\mathcal D$ are Alexandroff spaces. If the associated proset (preordered set) of the decomposition space $X/\mathcal D$ is a poset, then the decomposition map $\pi:X \to X/\mathcal D$ is a continuous map from the topological space $X$ to the poset $X/\mathcal D$ with the associated Alexandroff topology, which is nowadays called a poset-stratified space. As an application, we capture the face poset of a real hyperplane arrangement $\mathcal A$ of $\mathbb R^n$ as the associated poset of the decomposition space $\mathbb R^n/\mathcal D(\mathcal A)$ of the decomposition $\mathcal D(\mathcal A)$ determined by the arrangement $\mathcal A$. We also show that for any locally small category $\mathcal C$ the set $hom_{\mathcal C}(X,Y)$ of morphisms from $X$ to $Y$ can be considered as a poset-stratified space, and that for any objects $S, T$ (where $S$ plays as a source object and $T$ as a target object) there are a covariant functor $\mathfrak{st}^S_*: \mathcal C \to \mathcal Strat$ and a contravariant functor $\mathfrak{st}^*_T$ $\mathfrak{st}^*_T: \mathcal C \to \mathcal Strat$ from $\mathcal C$ to the category $\mathcal Strat$ of poset-stratified spaces. We also make a remark about Yoneda's Lemmas as to poset-stratified space structures of $hom_{\mathcal C}(X,Y)$.

Collective fixed points, generalized games and systems of generalized quasi-variational inclusion problems in topological spaces

Let ϕ : X → Y be an affine continuous surjection between compact convex sets. Suppose that the canonical copy of the space of real-valued affine continuous functions on Y in the space of real-valued affine continuous functions on X is complemented. We show that if F is a topological vector space, then f : Y → F is of affine Baire class α whenever the composition f ◦ ϕ is of affine Baire class α. This abstract result is applied to extend known results on affine Baire classes of strongly affine Baire mappings.

Completely normal frames and real-valued functions