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Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ + if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ +. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin...

We study maximal pseudocompact spaces calling them also MP-spaces. We show that the product of a maximal pseudocompact space and a countable compact space is maximal pseudocompact. If X is hereditarily maximal pseudocompact then X × Y is hereditarily maximal pseudocompact for any first countable compact space Y. It turns out that hereditary maximal pseudocompactness coincides with the Preiss-Simon property in countably compact spaces. In compact spaces, hereditary MP-property is invariant under...

A cardinal function $\varphi$ (or a property $𝒫$) 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 $𝒫$ ($\equiv X⊢𝒫$) iff $Y⊢𝒫$). 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 $f:X\to ℝ$. We will call a space $X$ exponentially separable if for any countable family $ℱ$ of closed subsets of $X$, there exists a countable set $A\subset X$ such that $A\cap \bigcap 𝒢\ne \varnothing$ whenever $𝒢\subset ℱ$ and $\bigcap 𝒢\ne \varnothing$. 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 $C_p\left(X\right)$. 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 ${𝐑}^{2n}$ but not necessarily over ${𝐑}^{2n-2}$. It is established that if a metrizable compact $X$ splits over ${𝐑}^n$, then $dimX\le n$. An example of $n$-dimensional compact space which does not split over ${𝐑}^{2n}$ 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 $\left|Y\right|\le 2^{𝔠}$ is scattered, then...

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have...

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 $𝒮$ of a space $X$, the game $PO(𝒮,X)$ 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 $U_n\in 𝒮$ such that $x_n\in U_n$. The game stops after the moves $\{x_n,U_n:n\in ø\}$ have been made and the player $P$ wins if ${\bigcup }_{n\in ø}{U}_n=X$; otherwise $O$ is the winner. Since $PO(𝒮,X)$ 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 $PO(𝒮,X)$ for a given subbase $𝒮$. It turns out that, for any subbase $𝒮$, the player $P$ has a winning strategy...

A in a space $X$ is a family $𝒪=\{O_x:x\in X\}$ of open subsets of $X$ such that $x\in O_x$ for any $x\in X$. A set $Y\subseteq X$ is if $𝒪\left(Y\right)=\bigcup \{O_x:x\in Y\}=X$. 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) $𝒫$, the class ${𝒫}^{*}$ dual to $𝒫$ (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 $Y\in 𝒫$ and $\bigcup \{O_x:x\in Y\}=X$. The spaces from ${𝒫}^{*}$ 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...