A cardinal function $\varphi $ (or a property $\mathcal{P}$) is called $l$-invariant if for any Tychonoff spaces $X$ and $Y$ with ${C}_{p}\left(X\right)$ and ${C}_{p}\left(Y\right)$ linearly homeomorphic we have $\varphi \left(X\right)=\varphi \left(Y\right)$ (or the space $X$ has $\mathcal{P}$ ($\equiv X\u22a2\mathcal{P}$) iff $Y\u22a2\mathcal{P}$). We prove that the hereditary Lindelöf number is $l$-invariant as well as that there are models of $ZFC$ in which hereditary separability is $l$-invariant.

We study systematically a class of spaces introduced by Sokolov and call them Sokolov spaces. Their importance can be seen from the fact that every Corson compact space is a Sokolov space. We show that every Sokolov space is collectionwise normal, $\omega $-stable and $\omega $-monolithic. It is also established that any Sokolov compact space $X$ is Fréchet-Urysohn and the space ${C}_{p}\left(X\right)$ is Lindelöf. We prove that any Sokolov space with a ${G}_{\delta}$-diagonal has a countable network and obtain some cardinality restrictions on subsets...

A space $X$ is functionally countable if $f\left(X\right)$ is countable for any continuous function $f:X\to \mathbb{R}$. We will call a space $X$ exponentially separable if for any countable family $\mathcal{F}$ of closed subsets of $X$, there exists a countable set $A\subset X$ such that $A\cap \bigcap \mathcal{G}\ne \varnothing $ whenever $\mathcal{G}\subset \mathcal{F}$ and $\bigcap \mathcal{G}\ne \varnothing $. Every exponentially separable space is functionally countable; we will show that for some nice classes of spaces exponential separability coincides with functional countability. We will also establish that the class of exponentially separable spaces has...

Given a Tychonoff space $X$ and an infinite cardinal $\kappa $, we prove that exponential $\kappa $-domination in $X$ is equivalent to exponential $\kappa $-cofinality of $\phantom{\rule{0.166667em}{0ex}}{C}_{p}\left(X\right)$. On the other hand, exponential $\kappa $-cofinality of $X$ is equivalent to exponential $\kappa $-domination in ${C}_{p}\left(X\right)$. We show that every exponentially $\kappa $-cofinal space $X$ has a ${\kappa}^{+}$-small diagonal; besides, if $X$ is $\kappa $-stable, then $nw\left(X\right)\le \kappa $. In particular, any compact exponentially $\kappa $-cofinal space has weight not exceeding $\kappa $. We also establish that any exponentially $\kappa $-cofinal space $X$ with...

The old game is the point-open one discovered independently by F. Galvin [7] and R. Telgársky [17]. Recall that it is played on a topological space $X$ as follows: at the $n$-th move the first player picks a point ${x}_{n}\in X$ and the second responds with choosing an open ${U}_{n}\ni {x}_{n}$. The game stops after $\omega $ moves and the first player wins if $\cup \{{U}_{n}:n\in \omega \}=X$. Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games $\theta $ and $\Omega $. In $\theta $ the moves are made exactly as in the point-open game, but the...

A space $X$ is splittable over a space $Y$ (or splits over $Y$) if for every $A\subset X$ there exists a continuous map $f:X\to Y$ with ${f}^{-1}fA=A$. We prove that any $n$-dimensional polyhedron splits over ${\mathbf{R}}^{2n}$ but not necessarily over ${\mathbf{R}}^{2n-2}$. It is established that if a metrizable compact $X$ splits over ${\mathbf{R}}^{n}$, then $dimX\le n$. An example of $n$-dimensional compact space which does not split over ${\mathbf{R}}^{2n}$ is given.

We show that exponential separability is an inverse invariant of closed maps with countably compact exponentially separable fibers. This implies that it is preserved by products with a scattered compact factor and in the products of sequential countably compact spaces. We also provide an example of a $\sigma $-compact crowded space in which all countable subspaces are scattered. If $X$ is a Lindelöf space and every $Y\subset X$ with $\left|Y\right|\le {2}^{{\omega}_{1}}$ is scattered, then $X$ is functionally countable; if every $Y\subset X$ with $\left|Y\right|\le {2}^{\U0001d520}$ is scattered, then...

In 2008 Juhász and Szentmiklóssy established that for every compact space $X$ there exists a discrete $D\subset X\times X$ with $\left|D\right|=d\left(X\right)$. We generalize this result in two directions: the first one is to prove that the same holds for any Lindelöf $\Sigma $-space $X$ and hence ${X}^{\omega}$ is $d$-separable. We give an example of a countably compact space $X$ such that ${X}^{\omega}$ is not $d$-separable. On the other hand, we show that for any Lindelöf $p$-space $X$ there exists a discrete subset $D\subset X\times X$ such that $\Delta =\{(x,x):x\in X\}\subset \overline{D}$; in particular, the diagonal $\Delta $ is a retract of $\overline{D}$ and the projection...

We prove that every countably compact AP-space is Fréchet-Urysohn. It is also established that if $X$ is a paracompact space and ${C}_{p}\left(X\right)$ is AP, then $X$ is a Hurewicz space. We show that every scattered space is WAP and give an example of a hereditarily WAP-space which is not an AP-space.

Given a subbase $\mathcal{S}$ of a space $X$, the game $PO(\mathcal{S},X)$ is defined for two players $P$ and $O$ who respectively pick, at the $n$-th move, a point ${x}_{n}\in X$ and a set ${U}_{n}\in \mathcal{S}$ such that ${x}_{n}\in {U}_{n}$. The game stops after the moves $\{{x}_{n},{U}_{n}:n\in \xf8\}$ have been made and the player $P$ wins if ${\bigcup}_{n\in \xf8}{U}_{n}=X$; otherwise $O$ is the winner. Since $PO(\mathcal{S},X)$ is an evident modification of the well-known point-open game $PO\left(X\right)$, the primary line of research is to describe the relationship between $PO\left(X\right)$ and $PO(\mathcal{S},X)$ for a given subbase $\mathcal{S}$. It turns out that, for any subbase $\mathcal{S}$, the player $P$ has a winning strategy...

A in a space $X$ is a family $\mathcal{O}=\{{O}_{x}:x\in X\}$ of open subsets of $X$ such that $x\in {O}_{x}$ for any $x\in X$. A set $Y\subseteq X$ is if $\mathcal{O}\left(Y\right)=\bigcup \{{O}_{x}:x\in Y\}=X$. If every neighbourhood assignment in $X$ has a closed and discrete (respectively, discrete) kernel, then $X$ is said to be a $D$-space (respectively a dually discrete space). In this paper we show among other things that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf $P$-space is a $D$-space and we prove an addition theorem for metalindelöf spaces which...

We prove that if ${X}^{n}$ is a union of $n$ subspaces of pointwise countable type then the space $X$ is of pointwise countable type. If ${X}^{\omega}$ is a countable union of ultracomplete spaces, the space ${X}^{\omega}$ is ultracomplete. We give, under CH, an example of a Čech-complete, countably compact and non-ultracomplete space, giving thus a partial answer to a question asked in [BY2].

Given a topological property (or a class) $\mathcal{P}$, the class ${\mathcal{P}}^{*}$ dual to $\mathcal{P}$ (with respect to neighbourhood assignments) consists of spaces $X$ such that for any neighbourhood assignment $\{{O}_{x}:x\in X\}$ there is $Y\subset X$ with $Y\in \mathcal{P}$ and $\bigcup \{{O}_{x}:x\in Y\}=X$. The spaces from ${\mathcal{P}}^{*}$ are called . We continue the study of this duality which constitutes a development of an idea of E. van Douwen used to define $D$-spaces. We prove a number of results on duals of some general classes of spaces establishing, in particular, that any generalized ordered space of countable...

It is shown that both the free topological group $F\left(X\right)$ and the free Abelian topological group $A\left(X\right)$ on a connected locally connected space $X$ are locally connected. For the Graev’s modification of the groups $F\left(X\right)$ and $A\left(X\right)$, the corresponding result is more symmetric: the groups $F\Gamma \left(X\right)$ and $A\Gamma \left(X\right)$ are connected and locally connected if $X$ is. However, the free (Abelian) totally bounded group $FTB\left(X\right)$ (resp., $ATB\left(X\right)$) is not locally connected no matter how “good” a space $X$ is. The above results imply that every non-trivial continuous homomorphism...

We study when a topological space has a weaker connected topology. Various sufficient and necessary conditions are given for a space to have a weaker Hausdorff or regular connected topology. It is proved that the property of a space of having a weaker Tychonoff topology is preserved by any of the free topological group functors. Examples are given for non-preservation of this property by “nice” continuous mappings. The requirement that a space have a weaker Tychonoff connected topology is rather...

We prove that it is independent of ZFC whether every Hausdorff countable space of weight less than $c$ has a dense regular subspace. Examples are given of countable Hausdorff spaces of weight $c$ which do not have dense Urysohn subspaces. We also construct an example of a countable Urysohn space, which has no dense completely Hausdorff subspace. On the other hand, we establish that every Hausdorff space of $\pi $-weight less than $\U0001d52d$ has a dense completely Hausdorff (and hence Urysohn) subspace. We show that...

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