We obtain some new properties of the class of KC-spaces, that is, those topological spaces in which compact sets are closed. The results are used to generalize theorems of Anderson [1] and Steiner and Steiner [12] concerning complementation in the lattice of ${T}_{1}$-topologies on a set $X$.

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...

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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