### On closed maps, increasing dimension

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It is established that a remainder of a non-locally compact topological group $G$ has the Baire property if and only if the space $G$ is not Čech-complete. We also show that if $G$ is a non-locally compact topological group of countable tightness, then either $G$ is submetrizable, or $G$ is the Čech-Stone remainder of an arbitrary remainder $Y$ of $G$. It follows that if $G$ and $H$ are non-submetrizable topological groups of countable tightness such that some remainders of $G$ and $H$ are homeomorphic, then the spaces...

Some strong versions of the Fréchet-Urysohn property are introduced and studied. We also strengthen the concept of countable tightness and generalize the notions of first-countability and countable base. A construction of a topological space is described which results, in particular, in a Tychonoff countable Fréchet-Urysohn space which is not first-countable at any point. It is shown that this space can be represented as the image of a countable metrizable space under a continuous pseudoopen mapping....

The class of $s$-spaces is studied in detail. It includes, in particular, all Čech-complete spaces, Lindelöf $p$-spaces, metrizable spaces with the weight $\le {2}^{\omega}$, but countable non-metrizable spaces and some metrizable spaces are not in it. It is shown that $s$-spaces are in a duality with Lindelöf $\Sigma $-spaces: $X$ is an $s$-space if and only if some (every) remainder of $X$ in a compactification is a Lindelöf $\Sigma $-space [Arhangel’skii A.V., Remainders of metrizable and close to metrizable spaces, Fund. Math. 220 (2013),...

We study topological spaces that can be represented as the union of a finite collection of dense metrizable subspaces. The assumption that the subspaces are dense in the union plays a crucial role below. In particular, Example 3.1 shows that a paracompact space $X$ which is the union of two dense metrizable subspaces need not be a $p$-space. However, if a normal space $X$ is the union of a finite family $\mu $ of dense subspaces each of which is metrizable by a complete metric, then $X$ is also metrizable by...

It is shown that there exists a $\sigma $-compact topological group which cannot be represented as a continuous image of a Lindelöf $p$-group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$-groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$-group, then there exists a covering $\gamma $ of $Y$ by dyadic compacta such that $\left|\gamma \right|\le {2}^{\omega}$. We also show that if a homogeneous compact space $Y$ is a continuous...

We prove a Dichotomy Theorem: for each Hausdorff compactification $bG$ of an arbitrary topological group $G$, the remainder $bG\setminus G$ is either pseudocompact or Lindelöf. It follows that if a remainder of a topological group is paracompact or Dieudonne complete, then the remainder is Lindelöf, and the group is a paracompact $p$-space. This answers a question in A.V. Arhangel’skii, , Moscow Univ. Math. Bull. 54:3 (1999), 1–6. It is shown that every Tychonoff space can be embedded as a closed subspace in a pseudocompact...

A well known theorem of W.W. Comfort and K.A. Ross, stating that every pseudocompact group is $C$-embedded in its Weil completion [5] (which is a compact group), is extended to some new classes of topological groups, and the proofs are very transparent, short and elementary (the key role in the proofs belongs to Lemmas 1.1 and 4.1). In particular, we introduce a new notion of canonical uniform tightness of a topological group $G$ and prove that every ${G}_{\delta}$-dense subspace $Y$ of a topological group $G$, such...

The following general question is considered. Suppose that $G$ is a topological group, and $F$, $M$ are subspaces of $G$ such that $G=MF$. Under these general assumptions, how are the properties of $F$ and $M$ related to the properties of $G$? For example, it is observed that if $M$ is closed metrizable and $F$ is compact, then $G$ is a paracompact $p$-space. Furthermore, if $M$ is closed and first countable, $F$ is a first countable compactum, and $FM=G$, then $G$ is also metrizable. Several other results of this kind are obtained....

Relative versions of many important theorems on cardinal invariants of topological spaces are formulated and proved on the basis of a general technical result, which provides an algorithm for such proofs. New relative cardinal invariants are defined, and open problems are discussed.

Starting with a very simple proof of Frol’ık’s theorem on homeomorphisms of extremally disconnected spaces, we show how this theorem implies a well known result of Malychin: that every extremally disconnected topological group contains an open and closed subgroup, consisting of elements of order $2$. We also apply Frol’ık’s theorem to obtain some further theorems on the structure of extremally disconnected topological groups and of semitopological groups with continuous inverse. In particular, every...

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...

Given a space $X$, its ${G}_{\delta}$-subsets form a basis of a new space ${X}_{\omega}$, called the ${G}_{\delta}$-modification of $X$. We study how the assumption that the ${G}_{\delta}$-modification ${X}_{\omega}$ is homogeneous influences properties of $X$. If $X$ is first countable, then ${X}_{\omega}$ is discrete and, hence, homogeneous. Thus, ${X}_{\omega}$ is much more often homogeneous than $X$ itself. We prove that if $X$ is a compact Hausdorff space of countable tightness such that the ${G}_{\delta}$-modification of $X$ is homogeneous, then the weight $w\left(X\right)$ of $X$ does not exceed ${2}^{\omega}$ (Theorem 1). We also establish...

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