Related papers: A Note On Separating Function Sets

We study point-separating function sets that are minimal with respect to the property of being separating. We first show that for a compact space $X$ having a minimal separating function set in $C_p(X)$ is equivalent to having a minimal…

We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff space $X$ equipped with the topology of pointwise convergence is distinguished if and only if $X$ is a $\Delta$-space in the sense of \cite…

We give new characterizations of spaces $X$ which are $k_\mathbb{R}$-spaces or $s_\mathbb{R}$-spaces. Applying the obtained results we provide some sufficient and necessary conditions on $X$ for which $C_p(X)$ is a $k_\mathbb{R}$-space or…

We study the complexity of the space $C^*_p(X)$ of bounded continuous functions with the topology of pointwise convergence. We are allowed to use descriptive set theoretical methods, since for a separable metrizable space $X$, the…

We give an example of an infinite metrizable space $X$ such that the space $C_p(X)$, of continuous real-valued function on $X$ endowed with the pointwise topology, is not homeomorphic to its own square $C_p(X)\times C_p(X)$. The space $X$…

In this paper we continue to study the property of separability of functional space C(X) with the open-point and bi-point-open topologies.

We give a condition under which a divisor X in a bounded convex domain of finite type D in C^n is the zero set of a function in a Hardy space H^p(D) for some p \textgreater{} 0. This generalizes Varopoulos' result [Zero sets of H^p…

We study properties of strongly separately continuous mappings defined on subsets of products of topological spaces equipped with the topology of pointwise convergence. In particular, we give a necessary and sufficient condition for a…

In this paper we consider the question of when the space $C_p(X)$ of continuous real-valued functions on $X$ with the pointwise convergence topology is countable dense homogeneous. In particular, we focus on the case when $X$ is countable…

It is known that for $X$ a nowhere locally compact metric space, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on $X$ contains a dense $G_\delta$ set in the space $C_b(X)$ of all bounded…

Let $X$ be metrizable, $Y$ be perfectly normal and suppose that there exists a uniformly continuous surjection $T: C_{p}(X) \to C_{p}(Y)$ (resp., $T: C_{p}^*(X) \to C_{p}^*(Y)$), where $C_{p}(X)$ (resp., $C_{p}^*(X)$) denotes the space of…

It is investigated the existence of a separately continuous function $f:X\times Y\to \mathbb R$ with an onepoint set of discontinuity for topological spaces $X$ and $Y$ which satisfy compactness type conditions. In particular, it is shown…

We give a characterization of countable discrete subspace $A$ of a topological space $X$ such that there exists a (linear) continuous mapping $\varphi:C_p^*(A)\to C_p(X)$ with $\varphi(y)|_A=y$ for every $y\in C_p^*(A)$. Using this…

As proved in [16], for a Tychonoff space $X$, a locally convex space $C_{p}(X)$ is distinguished if and only if $X$ is a $\Delta$-space. If there exists a linear continuous surjective mapping $T:C_p(X) \to C_p(Y)$ and $C_p(X)$ is…

For a Tychonoff space $X$, we will denote by $USC_{p}(X)$ ($B_1(X)$) a set of all real-valued upper semicontinuous functions (a set of all Baire functions of class 1) defined on $X$ endowed with the pointwise convergence topology. In this…

It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $\Lambda$. We show that the same is true for much wider spaces of continuous functions. In particular,…

We give some sufficient conditions of separation of two sets of integer points by a hyperplane. Our conditions are related to the notion of convexity of sets of integer points and are weaker than existing notions.

In this paper a construction of a metrizable zero-dimensional CDH space $X$ such that $X^2$ has exactly $\mathfrak{c}$ countable dense subsets is provided. Furthermore, it is shown that the space can be constructed consistently co-analytic.…

In this paper we study the property of separability of functional space with the open-point and bi-point-open topologies.

In this paper we consider some properties of a space B(X) of Borel functions on a set of reals X, with pointwise topology, that are stronger than separability.