Related papers: Lusin and Suslin properties of function spaces

Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. We show that the following assertions are equivalent: (i) $L(X)$ is $\ell_\infty$-barrelled, (ii) $L(X)$ is $\ell_\infty$-quasibarrelled, (iii) $L(X)$ is…

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,…

To every log-concave function $f$ one may associate a pair of measures $(\mu_{f},\nu_{f})$ which are the surface area measures of $f$. These are a functional extension of the classical surface area measure of a convex body, and measure how…

We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used. We study the relations of such…

A topological space $X$ is called Piotrowski if every quasicontinuous map $f:Z\to X$ from a Baire space $Z$ to $X$ has a continuity point. In this paper we survey known results on Piotrowski spaces and investigate the relation of Piotrowski…

In the present paper, we prove that a topological space admits a functorial Lindel\"ofification if and only if its realcompactification is Lindel\"of. To investigate the functorial Lindel\"ofifiability of a topological space, for each…

A space X is finite dimensional, locally compact and separable metrizable if and only if X has a finite basic family: continuous functions Phi_1,...,Phi_n of X to the reals, R, such that for all continuous f from X to R there are g_1,...,…

For a signature L with at least one constant symbol, an L-structure is called minimal if it has no proper substructures. Let S_L be the set of isomorphism types of minimal L-structures. The elements of S_L can be identified with…

Suppose G is a topological group containing a (closed) topological copy of the Frechet-Urysohn fan. If G is a perfectly normal sequential space (a normal k-space) then every closed metrizable subset in $G$ is locally compact. Applying this…

A topological space is almost locally compact if it contains a dense locally compact subspace. We generalize a result from \cite{Ma}, showing that isomorphism on Borel classes of almost locally compact Polish metric structures is always…

Let $X$ be a space of homogeneous type. Assume that $L$ is an non-negative second-order self-adjoint operator on $L^2\left(X\right)$ with (heart) kernel associated to the semigroup $e^{ - tL}$ that satisfies the Gaussian upper bound. In…

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…

A sequence of functions f_n: X -> R from a Baire space X to the reals is said to converge in category iff every subsequence has a subsequence which converges on all but a meager set. We show that if there exists a Souslin Tree then there…

For a set $X\sbst\R$, let $B(X)\sbst\R^X$ denote the space of Borel real-valued functions on $X$, with the topology inherited from the Tychonoff product $\R^X$. Assume that for each countable $A\sbst B(X)$, each $f$ in the closure of $A$ is…

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…

For a compact space $K$ we denote by $C_w(K)$ ($C_p(K)$) the space of continuous real-valued functions on $K$ endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose…

We consider approximations of a continuous function on a countable normed Fr\'{e}chet space by analytic and $*$-analytic. Also we found a criterium of the existence of an extension of a continuous function from a dense subspace of a…

Denote by $\mathbf C_k[\mathfrak M]$ the $C_k$-stable closure of the class $\mathfrak M$ of all metrizable spaces, i.e., $\mathbf C_k[\mathfrak M]$ is the smallest class of topological spaces that contains $\mathfrak M$ and is closed under…

We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number $k$ of real continuous functions $f_1,\cdots, f_k$ such that the functions $f_i\circ T^n$, $n\in\mathbb Z$, $1\leq i\leq k,$ span a…

A topological space $X$ is a $\Delta$-space (or $X \in \Delta$) if for any decreasing sequence $\{A_n : n < \omega\}$ of subsets of $X$ with empty intersection there is a (decreasing) sequence $\{U_n : n < \omega\}$ of open sets with empty…