Related papers: Baire property of some function spaces

Let $(X,\tau)$ be a Hausdorff space, where $X$ is an infinite set. The compact complement topology $\tau^{\star}$ on $X$ is defined by: $\tau^{\star}=\{\emptyset\} \cup \{X\setminus M, \text{where $M$ is compact in $(X,\tau)$}\}$. In this…

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in G$…

We introduce the natural and fairly general notion of a subanalytic bundle (with a finite dimensional vector space $P$ of sections) on a subanalytic subset $X$ of a real analytic manifold $M$, and prove that when $M$ is compact, there is a…

Let $(\mathcal{X}, \rho, \mu)$ be a metric measure space of homogeneous type which supports a certain Poincar\'e inequality. Denote by the symbol $\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$ the space of all continuous functions $f$ with…

Let X be a separable metric space and let \beta be the strict topology on the space of bounded continuous functions on X, which has the space of \tau-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonne\'{e} type…

Kechris and Louveau showed that each real-valued bounded Baire class 1 function defined on a compact metric space can be written as an alternating sum of a decreasing countable transfinite sequence of upper semi-continuous functions.…

A topological space is nonseparably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected sequential topological space X is the image of a nonseparably connected complete…

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps f:X \to G endowed with the Whitney (graph) topology and by C_c(X,G) the subgroup consisting of…

For any Tychonoff space $X$ let $D(X)$ be either the set $C(X)$ of all continuous functions on $X$ or the set $C^*(X)$ of all bounded continuous functions on $X$. When $D(X)$ is endowed with the point convergence topology, we write…

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 prove that if $X$ is a Polish space and $F$ is a face of $P(X)$ with the Baire property, then $F$ is either a meager or a co-meager subset of $P(X)$. As a consequence we show that for every abelian Polish group $X$ and every analytic…

A space $X$ is said to be $\pi$-metrizable if it has a $\sigma$-discrete $\pi$-base. In this paper, we mainly give affirmative answers for two questions about $\pi$-metrizable spaces. The main results are that: (1) A space $X$ is…

Denote by $\mathbf C_p[\mathfrak M_0]$ the $C_p$-stable closure of the class $\mathfrak M_0$ of all separable metrizable spaces, i.e., $\mathbf C_p[\mathfrak M_0]$ is the smallest class of topological spaces that contains $\mathfrak M_0$…

We prove that for any topological space $X$ of countable tightness, each \sigma-convex subspace $\F$ of the space $SC_p(X)$ of scatteredly continuous real-valued functions on $X$ has network weight $nw(\F)\le nw(X)$. This implies that for a…

We show that all sufficiently nice $\lambda$-sets are countable dense homogeneous ($\mathsf{CDH}$). From this fact we conclude that for every uncountable cardinal $\kappa \le \mathfrak{b}$ there is a countable dense homogeneous metric space…

We answer two questions from {\it V.Bykov, On Baire class one functions on a product space, Topol. Appl. {199} (2016) 55--62,} and prove that every Baire one function on a subspace of a countable perfectly normal product is the pointwise…

A Tychonoff space $X$ is called ({\em sequentially}) {\em Ascoli} if every compact subset (resp. convergent sequence) of $C_k(X)$ is equicontinuous, where $C_k(X)$ denotes the space of all real-valued continuous functions on $X$ endowed…

For any Tychonoff space $X$ let $C_p(X)$ (resp., $C^*_p(X)$) be the set of all continuous (resp., and bounded) functions on $X$ with the pointwise convergence topology. Given Tychonoff spaces $X$ and $Y$, Uspenskij \cite{us} proved that if…

We prove that an asymmetric normed space is never a Baire space if the topology induced by the asymmetric norm is not equivalent to the topology of a norm. More precisely, we show that a biBanach asymmetric normed space is a Baire space if…

Let M be the countably infinite metric fan. We show that C_k(M,2) is sequential and contains a closed copy of Arens space S_2. It follows that if X is metrizable but not locally compact, then C_k(X) contains a closed copy of S_2, and hence…