We consider the class of compact spaces $K_A$ which are modifications of the well known double arrow space. The space $K_A$ is obtained from a closed subset K of the unit interval [0,1] by “splitting” points from a subset A ⊂ K. The class of all such spaces coincides with the class of separable linearly ordered compact spaces. We prove some results on the topological classification of $K_A$ spaces and on the isomorphic classification of the Banach spaces $C\left(K_A\right)$.

We show that there exists an Abelian topological group $G$ such that the operations in $G$ cannot be extended to the Dieudonné completion $\mu G$ of the space $G$ in such a way that $G$ becomes a topological subgroup of the topological group $\mu G$. This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the...

For Tychonoff $X$ and $\alpha$ an infinite cardinal, let $\alpha defX:=$ the minimum number of $\alpha$ cozero-sets of the Čech-Stone compactification which intersect to $X$ (generalizing $ℝ$-defect), and let $rtX:={min}_{\alpha }max(\alpha ,\alpha defX)$. Give $C\left(X\right)$ the compact-open topology. It is shown that $\tau C\left(X\right)\le n\chi C\left(X\right)\le rtX=max\left(L\right(X),L(X)defX)$, where: $\tau$ is tightness; $n\chi$ is the network character; $L\left(X\right)$ is the Lindel"of number. For example, it follows that, for $X$ Čech-complete, $\tau C\left(X\right)=L\left(X\right)$. The (apparently new) cardinal functions $n\chi C$ and $rt$ are compared with several others.

Sibley and Sempi have constructed metrics on the space of probability distribution functions with the property that weak convergence of a sequence is equivalent to metric convergence. Sibley's work is a modification of Levy's metric, but Sempi's construction is of a different sort. Here we construct a family of metrics having the same convergence properties as Sibley's and Sempi's but which does not appear to be related to theirs in any simple way. Some instances are brought out in which the metrics...

Let $X$ be a Baire space, $Y$ be a compact Hausdorff space and $\varphi :X\to C_p\left(Y\right)$ be a quasi-continuous mapping. For a proximal subset $H$ of $Y\times Y$ we will use topological games ${𝒢}_1\left(H\right)$ and ${𝒢}_2\left(H\right)$ on $Y\times Y$ between two players to prove that if the first player has a winning strategy in these games, then $\varphi$ is norm continuous on a dense $G_{\delta }$ subset of $X$. It follows that if $Y$ is Valdivia compact, each quasi-continuous mapping from a Baire space $X$ to $C_p\left(Y\right)$ is norm continuous on a dense $G_{\delta }$ subset of $X$.

Let be the class of Banach spaces X for which every weakly quasi-continuous mapping f: A → X defined on an α-favorable space A is norm continuous at the points of a dense $G_{\delta }$ subset of A. We will show that this class is stable under c₀-sums and ${\ell }^p$-sums of Banach spaces for 1 ≤ p < ∞.

We prove that a cosmic space (= a Tychonoff space with a countable network) is analytic if it is an image of a $K$-analytic space under a measurable mapping. We also obtain characterizations of analyticity and $\sigma$-compactness in cosmic spaces in terms of metrizable continuous images. As an application, we show that if $X$ is a separable metrizable space and $Y$ is its dense subspace then the space of restricted continuous functions $C_p(X\mid Y)$ is analytic iff it is a $K_{\sigma \delta }$-space iff $X$ is $\sigma$-compact.

Several remarks on the properties of approximation by points (AP) and weak approximation by points (WAP) are presented. We look in particular at their behavior in product and at their relationships with radiality, pseudoradiality and related concepts. For instance, relevant facts are: (a) There is in ZFC a product of a countable WAP space with a convergent sequence which fails to be WAP. (b) $C_p$ over $\sigma$-compact space is AP. Therefore AP does not imply even pseudoradiality in function spaces, while...

We characterize Peano continua using Bing-Krasinkiewicz-Lelek maps. Also we deal with some topics on Whitney preserving maps.