Equiconnected Space Research Articles

Comparison of some set open and uniform topologies and some properties of the restriction map

Let $X$ be a Tychonoff space, $Y$ an equiconnected space and $C(X,Y)$ be the set of all continuous functions from $X$ to $Y$. In this paper, we provide a criterion for the coincidence of set open and uniform topologies on $C(X,Y)$ when these topologies are defined by a family $\alpha$ consisting of $Y$-compact subsets of $X$. For a subspace $Z$ of a topological space $X$, we also study the continuity and the openness of the restriction map $\pi_{Z}:C(X,Y)\rightarrow C(Z,Y)$ when both $C(X,Y)$ and $C(Z,Y)$ are endowed with the set open topology.

Diagonals of separately continuous functions of $n$ variables with values in strongly $\sigma$-metrizable spaces

We prove the result on Baire classification of mappings $f:X\times Y\to Z$ which are continuous with respect to the first variable and belongs to a Baire class with respect to the second one, where $X$ is a $PP$-space, $Y$ is a topological space and $Z$ is a strongly $\sigma$-metrizable space with additional properties. We show that for any topological space $X$, special equiconnected space $Z$ and a mapping $g:X\to Z$ of the $(n-1)$-th Baire class there exists a strongly separately continuous mapping $f:X^n\to Z$ with the diagonal $g$. For wide classes of spaces $X$ and $Z$ we prove that diagonals of separately continuous mappings $f:X^n\to Z$ are exactly the functions of the $(n-1)$-th Baire class. An example of equiconnected space $Z$ and a Baire-one mapping $g:[0,1]\to Z$, which is not a diagonal of any separately continuous mapping $f:[0,1]^2\to Z$, is constructed.

Diagonals of separately continuous functions and their analogs

We prove that for a topological space X, an equiconnected space Z and a Baire-one mapping g:X→Z there exists a separately continuous mapping f:X2→Z with the diagonal g, i.e. g(x)=f(x,x) for every x∈X. Under a mild assumptions on X and Z we obtain that diagonals of separately continuous mappings f:X2→Z are exactly Baire-one functions, and diagonals of mappings f:X2→Z which are continuous on the first variable and Lipschitz (differentiable) on the second one, are exactly the functions of stable first Baire class.

Equiconnected spaces and Baire classification of separately continuous functions and their analogs

Abstract We investigate the Baire classification of mappings f: X × Y → Z, where X belongs to a wide class of spaces which includes all metrizable spaces, Y is a topological space, Z is an equiconnected space, which are continuous in the first variable. We show that for a dense set in X these mappings are functions of a Baire class α in the second variable.

Some upper bounds for density of function spaces

Let C α ( X , Y ) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X which is closed under finite unions. We proved that the density of the space C α ( X , Y ) is at most iw ( X ) ⋅ d ( Y ) where iw ( X ) denotes the i-weight of the Tychonoff space X, and d ( Y ) denotes the density of the space Y when Y is an equiconnected space with equiconnecting function Ψ, and Y has a base consists of Ψ-convex subsets of Y. We also prove that the equiconnectedness of the space Y cannot be replaced with pathwise connectedness of Y. In fact, it is shown that for each infinite cardinal κ, there is a pathwise connected space Y such that π-weight of Y is κ, but Souslin number of the space C k ( [ 0 , 1 ] , Y ) is 2 κ .

Some cardinal invariants on the space [formula omitted

Let C α ( X , Y ) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X such that closed under finite unions. We define two properties ( E 1 ) and ( E 2 ) on the triple ( α , X , Y ) which yield new equalities and inequalities between some cardinal invariants on C α ( X , Y ) and some cardinal invariants on the spaces X, Y such as: Theorem If Y is an equiconnected space with a base consisting of φ-convex sets, then for each f ∈ C ( X , Y ) , χ ( f , C α ( X , Y ) ) = α a ( X ) . w e ( f ( X ) ) . Corollary Let Y be a noncompact metric space and let the triple ( α , X , Y ) satisfy ( E 1 ) . The following are equivalent: (i) C α ( X , Y ) is a first-countable space. (ii) π-character of the space C α ( X , Y ) is countable. (iii) C α ( X , Y ) is of pointwise countable type. (iv) There exists a compact subset K of C α ( X , Y ) such that π-character of K in the space C α ( X , Y ) is countable. (v) α a ( X ) ⩽ ℵ 0 . (vi) C α ( X , Y ) is metrizable. (vii) C α ( X , Y ) is a q-space. (viii) There exists a sequence { O n : n ∈ ω } of nonempty open subset of C α ( X , Y ) such that each sequence { g n : n ∈ ω } with g n ∈ O n for each n ∈ ω , has a cluster point in C α ( X , Y ) .

Convergence of mann iteration processes for nonexpansive operators in equi-connected metric spaces

The paper deals with various conditions implying the convergence of a Mann type iteration process constructed for a non-expansive operator in an equi-connected space (i. e. metric space equipped with a connecting function; so the iterates are taken along certain curves). Coefficients of the iterates do not have to be separated from 0 or 1. 1].

Retracts in equiconnected spaces

The object of this paper is to show that the concepts “retract” and “strong deformation retract” coincide for a subset of an equiconnected space. Also, we have a similar local version in a locally equiconnected space.

Retracts in metric spaces

In this paper we define S-contractibility and two classes of spaces connected with this notion. A space X is said to be S-contractible provided that S is a function S : X × ⟨ 0 , 1 ⟩ × X ( x , α , y ) ↦ S x ( α , y ) ∈ X S:X \times \langle 0,1\rangle \times X (x,\alpha ,y) \mapsto {S_x}(\alpha ,y) \in X that is continuous in α \alpha and y, and for every x , y ∈ X , S x ( 0 , y ) = y , S x ( 1 , y ) = x x,y \in X,{S_x}(0,y) = y,{S_x}(1,y) = x . This notion is close to equiconnectedness, which can be defined as follows. A space X is equiconnected if there exists a map S such that X is S-contractible and S x ( α , x ) = x {S_x}(\alpha ,x) = x for all x ∈ X x \in X and α ∈ I \alpha \in I (cf. [4]). The results we obtain in the theory of retracts are close to those that are known for equiconnected spaces. Also the thickness of the neighborhood that can be retracted on a set in a metric space is estimated, which enables to prove a theorem belonging to fixed point theory.