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We show that if an uncountable regular cardinal τ and τ + 1 embed in a topological group G as closed subspaces then G is not normal. We also prove that an uncountable regular cardinal cannot be embedded in a torsion free Abelian group that is hereditarily normal. These results are corollaries to our main results about ordinals in topological groups. To state the main results, let τ be an uncountable regular cardinal and G a T₁ topological group. We prove, among others, the following statements:...

We investigate spaces $C_p(·,n)$ over LOTS (linearly ordered topological spaces). We find natural necessary conditions for linear Lindelöfness of $C_p(·,n)$ over LOTS. We also characterize countably compact LOTS whose $C_p(·,n)$ is linearly Lindelöf for each n. Both the necessary conditions and the characterization are given in terms of the topology of the Dedekind completion of a LOTS.

It is shown that every strong $\Sigma$ space is a $D$-space. In particular, it follows that every paracompact $\Sigma$ space is a $D$-space.

We investigate how the Lindelöf property of the function space $C_p(X,Y)$ is influenced by slight changes in $X$ and/or $Y$.

We show that if $X$ is first-countable, of countable extent, and a subspace of some ordinal, then $C_p(X,2)$ is Lindelöf.

The results concern clopen sets in products of topological spaces. It is shown that a clopen subset of the product of two separable metrizable (or locally compact) spaces is not always a union of clopen boxes. It is also proved that any clopen subset of the product of two spaces, one of which is compact, can always be represented as a union of clopen boxes.

We show that if $X^2$ has countable extent and $X$ has a zeroset diagonal then $X$ is submetrizable. We also make a couple of observations regarding spaces with a regular $G_{\delta }$-diagonal.

We introduce a notion of absolute submetrizability (= ``every Tychonoff subtopology is submetrizable'') and investigate its behavior under basic topological operations. The main result is an example of an absolutely submetrizable space that contains an uncountable set of isolated points (hence the space is neither separable nor hereditarily Lindelöf). This example is used to show that absolute submetrizability is not preserved by some topological operations, in particular, by free sums.

It is shown that if $X$ is a first-countable countably compact subspace of ordinals then $C_p\left(X\right)$ is Lindelöf. This result is used to construct an example of a countably compact space $X$ such that the extent of $C_p\left(X\right)$ is less than the Lindelöf number of $C_p\left(X\right)$. This example answers negatively Reznichenko’s question whether Baturov’s theorem holds for countably compact spaces.

It is proved that if a regular space $X$ is the union of a finite family of metrizable subspaces then $X$ is a $D$-space in the sense of E. van Douwen. It follows that if a regular space $X$ of countable extent is the union of a finite collection of metrizable subspaces then $X$ is Lindelöf. The proofs are based on a principal result of this paper: every space with a point-countable base is a $D$-space. Some other new results on the properties of spaces which are unions of a finite collection of nice subspaces...

Several topological properties lying between the submetrizability and the $G_{\delta }$-diagonal property are studied. We are mostly interested in their relationship to each other and to the submetrizability. The first example of a Tychonoff space with a regular $G_{\delta }$-diagonal but without a zero-set diagonal is given. The same example shows that a Tychonoff separable space with a regular $G_{\delta }$-diagonal need not be submetrizable. We give a necessary and sufficient condition for submetrizability of a regular separable...

Let $\tau$ be an uncountable regular cardinal and $G$ a $T_1$ topological group. We prove the following statements: (1) If $\tau$ is homeomorphic to a closed subspace of $G$, $G$ is Abelian, and the order of every non-neutral element of $G$ is greater than $5$ then $\tau \times \tau$ embeds in $G$ as a closed subspace. (2) If $G$ is Abelian, algebraically generated by $\tau \subset G$, and the order of every element does not exceed $3$ then $\tau \times \tau$ is not embeddable in $G$. (3) There exists an Abelian topological group $H$ such that ${\omega }_1$ is homeomorphic to a closed subspace...

Let $X$ be a compact Hausdorff space with a point $x$ such that $X\setminus \left\{x\right\}$ is linearly Lindelöf. Is then $X$ first countable at $x$? What if this is true for every $x$ in $X$? We consider these and some related questions, and obtain partial answers; in particular, we prove that the answer to the second question is “yes” when $X$ is, in addition, $\omega$-monolithic. We also prove that if $X$ is compact, Hausdorff, and $X\setminus \left\{x\right\}$ is strongly discretely Lindelöf, for every $x$ in $X$, then $X$ is first countable. An example of linearly Lindelöf...

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...