Spaces on which every Continuous Map into a given Space is Constant

Chapter 9 - Concrete Categories

Bifurcation Theory and Periodic Solutions of Some Autonomous Functional Differential Equations

Functorial admissible quasi-uniformities on topological spaces

In this paper, we investigate the preservation of some topological properties during the weakly discontinuous, scatteredly continuous, and point-wise discontinuous maps on topological spaces. In addition, we also investigate the preservation of topological properties during such corresponding homeomorphisms. Furthermore, we also establish some significant relationships among weakly discontinuous, scatteredly continuous, and point-wise discontinuous maps. Scatteredly continuous map, weakly discontinuous map, point-wise discontinuous map, weak homeomorphism, scattered homeomorphism, point-wise homeomorphism.

c-16 - Corson Compacta

Assuming the continuum hypothesis, a set X of real numbers will be constructed such that every compact continuous image of X is countable, but XXX admits a uniformly continuous mapping onto the Cantor set. This completes (modulo the continuum hypothesis and the Ulam measure problem) the determination of whether a product or coproduct of m Boolean algebras or fields of sets can have an infinite free, or projective, or injective subor quotient object when the factors do not [1]. The concluding construction (modulo the hypothesis) is embarrassingly easy, but the result is unlike the others [1 ]; this is the only instance of a finite coproduct creating a remarkable subobject (viz. an infinite free subfield), and there is no instance among these of a finite product creating a remarkable quotient. The use made of the continuum hypothesis seems slight. However, Sierpinski has pointed out that the weaker result that there is a set of real numbers of the power of the continuum admitting no continuous mapping onto [0, 1] (C5 in [2]) has not been established without the hypothesis. On the continuum hypothesis, that is a by-product of an important construction of Lusin, the subject of Chapter II of [2]. We shall find X in the Cantor set C considered as an infinite power of the compact group Z2 (written additively). By a lemma of Lavrentiev (Theorem 99 of [3]), any continuous mapping of a subset of C onto [0, 1] can be extended over a Ga subset of C. There are only c (the power of the continuum) Ga sets, and each has only c continuous mappings to [0, 1]. Hence we can index all these mappingsfa by ordinals of smaller cardinal than c, and index similarly the points xa of C. For each fa, the inverse images of points are c disjoint subsets of C each closed in their union; so except for countably many, they are nowhere dense. By the continuum hypothesis, C is not a union of fewer than c such sets. There is no difficulty in building up sets S, T, no subset of either of which is mapped onto [0, 1] by any fa, but with every xa representable as sa+ta, Sa in S and ta in T. Carry along expanding sets Ha disjoint from S, Ka disjoint from T (Ho and Ko empty). Arrived

Assume that $X$ is a non-empty set and $T$ and $S$ are real or complex mappings defined on the product $X \times X$. Additive and multiplicative Sincov's equations are: $$T(x,z) = T(x, y ) + T(y, z)$$ and $$S(x,z) = S(x, y ) \cdot S(y, z),$$ respectively. Both equations play important roles in many areas of mathematics. In the present paper we study related inequalities. We deal with functional inequality $$ G(x,z) \leq G(x, y ) \cdot G(y, z), \quad x , y, z \in X $$ and we assume that $X$ is a topological space and $G\colon X \times X \to \mathbb{R}$ is a continuous mapping. In some our statements a considerably weaker regularity than continuity of $G$ is needed. We also study the reverse inequality: $$F(x,z) \geq F(x, y ) \cdot F(y, z), \quad x , y, z \in X $$ and the additive inequality (the triangle inequality): $$H(x,z) \leq H(x,y) + H(y,z), \quad x, y , z \in X.$$ A corollary for generalized (non-symmetric) metric is derived.

The main objective of this paper is to introduce a new generalized form of topological vector spaces, namely, almost $\\alpha$-topological vector spaces by using the concepts of $\\alpha$-open sets, regular open sets, and almost $\\alpha$-continuous mappings. In the present paper, some important examples of almost $\\alpha$-topological vector spaces with basic properties are characterized.

The concept of fuzzy Fréchet space Z introduced in two equivalent ways, the first way when Z considered as a fuzzy F − locally convex topological vector space and the second one when Z defined as a complete fuzzy topological vector space whose fuzzy topology generated by a countable separating collection of fuzzy seminorms. Moreover, many classes of continuous mappings between these spaces are established and specified that the class of continuous, strongly continuous and sequentially continuous mappings between these spaces are equivalent.

We consider continuous mappings between two Banach spaces that depend on a parameter with values in a topological space. These mappings are assumed to be continuously differentiable for each value of the parameter. Under normality (regularity) assumptions of the mappings under consideration, we obtain sufficient conditions for the existence of global and semilocal implicit functions. A priori estimates for solutions are given. As an application of these results, we obtain, in particular, a theorem on extending an implicit function from a given closed set to the whole parameter space and a theorem on coincidence points of mappings. Bibliography: 32 titles.

Among the twenty-three mathematical problems that Hilbert posed at the Second International Congress of Mathematicians in Paris in 1900 (see [10]), the fifth problem is concerned with Lie’s theory of transformation groups, and in its second part deals with what Hilbert calls “infinite groups,” but which are not groups in the modern use of the term. The questions in this second part of the fifth problem concern functional equations and difference equations and have, for example, connections with the work of N.H. Abel. These questions lie completely outside the theory of transformation groups, and we do not discuss them here any further. We refer the reader to [1] for the “state of the art” of 1989 in this area of research. We recall that a topological transformation group consists of a topological group G, a topological space X, and a continuous action of G on X, that is, a continuous map

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 $hom_{\mathcal C}(X,Y)$.

A proof of topological completeness for S4 in (0, 1)

We prove that, given a topological space \(X\), the following conditions are equivalent. (\(\upalpha \)) \(X\) is a Gruenhage space. (\(\upbeta \)) \(X\) has a countable cover by sets of small local diameter (property SLD) by \({{\mathcal F}}\cap {{\mathcal G}}\) sets. (\(\upgamma \)) \(X\) has a separating \(\sigma \)-isolated family \({{\mathcal M}}\subset {{\mathcal F}}\cap {{\mathcal G}}\). (\(\updelta \)) \(X\) has a one-to-one continuous map into a metric space which has a \(\sigma \)-isolated base of \({{\mathcal F}}\cap {{\mathcal G}}\) sets. Besides, we provide an example which shows $$\begin{aligned} \text{ Fragmentability } \nRightarrow \text{ property } \text{ SLD } \nRightarrow \text{ the } \text{ space } \text{ to } \text{ be } \text{ Gruenhage. } \end{aligned}$$

In this paper, we introduce the concept of αγ-open sets as a generalization of γ-open sets in a topological space (X, τ). Using this set, we introduce αγT0, αγ-T½, αγT1, αγT2, αγD0, αγD1 and αγD2 spaces and study some of its properties. Finally we introduce α(γ,γ')-continuous mappings and give some properties of such mappings.