arXiv subjects
Publications and source records attributed to Ol'ga Sipacheva.
Let $K$ be either a complete nondiscrete second-countable valued field (e.g., $\mathbb R$ or $\mathbb C$) or a finite field. It is proved that any countable-dimensional topological vector space over $K$ has a closed discrete basis.
It is proved that any infinite elementary Abelian topological group has an infinite maximally almost periodic subgroup.
A topological space $X$ is $\mathbb R^{ω_1}$-factorizable if any continuous function $f\colon X\to \mathbb R^{ω_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$ is $\mathbb R^{ω_1}$-factorizable if and only if $X\times D(ω_1)$, where $D(ω_1)$ is a discrete space of cardinality $ω_1$, is $z$-embedded in the product $βX\times βD(ω_1)$ of the Stone--Cech compactifications. It is also proved that $\mathbb R^{ω_1}$-factorizability is hereditary and countably multiplicative, that any $\mathbb R^{ω_1}$-factorizable space is hereditarily Lindelöf and hereditarily separable, and that the existence of nonmetrizable $\mathbb R^{ω_1}$-factorizable topological spaces and groups is independent of ZFC: under CH, all $\mathbb R^{ω_1}$-factorizable spaces are second-countable, while under MA + $\lnot$CH, the countable Fréchet--Urysohn fan is $\mathbb R^{ω_1}$-factorizable.
Strongly zero-dimensional topological groups $G_1$, $G_2$, and $G$ such that $G_1\times G_2$ has positive covering dimension and $G$ contains a closed subgroup of positive covering dimension are constructed. Moreover, all finite powers of $G_1$ are Lindelöf and $G_2$ is second-countable. An example of a strongly zero-dimensional space $X$ whose free, free Abelian, and free Boolean topological groups have positive covering dimension is also given.
An example of two $\mathbb R$-factorizable groups whose product is not $\mathbb R$-factorizable is constructed. One of these groups is second-countable and the other Lindelöf to any finite power.
The problem of the existence of non-pseudo-$\aleph_1$-compact $\mathbb R$-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than $ω_1$. Closely related results concerning the $\mathbb R$-factorizability of products of topological groups and spaces are also obtained (a product $X\times Y$ of topological spaces is said to be $\mathbb R$-factorizable if any continuous function $X\times Y\to \mathbb R$ factors through a product of maps from $X$ and $Y$ to second-countable spaces). In particular, it is proved that the square $G\times G$ of a topological groups $G$ is $\mathbb R$-factorizable as a group if and only if it is $\mathbb R$-factorizable as a product of spaces, in which case $G$ is pseudo-$\aleph_1$-compact. It is also proved that if the product of a space $X$ and an uncountable discrete space is $\mathbb R$-factorizable, then $X^ω$ is heredirarily separable and heredirarily Lindelöf.
A strongly zero-dimensional topological group containing a closed subgroup of positive covering dimension is constructed.
An ultrafilter $p$ on $ω$ is said to be discrete if, given any function $f\colon ω\to X$ to any completely regular Hausdorff space, there is an $A \in p$ such that $f(A)$ is discrete. Basic properties of discrete ultrafilters are studied. Three intermediate classes of spaces $\mathscr R_1 \subset \mathscr R_2 \subset \mathscr R_3$ between the class of $F$-spaces and the class of van~Douwen's $βω$-spaces are introduced. It is proved that no product of infinite compact $\mathscr R_2$-spaces is homogeneous; moreover, under the assumption $\mathfrak d =\mathfrak c$, no product of $βω$-spaces is homogeneous.
Topological groups whose underlying spaces are basically disconnected, $F$-, or $F'$-spaces but not $P$-spaces are considered. It is proved, in particular, that the existence of a Lindelöf basically disconnected topological group which is not a $P$-space is equivalent to the existence of a Boolean basically disconnected Lindelöf group of countable pseudocharacter, that free and free Abelian topological groups of zero-dimensional non-$P$-spaces are never $F'$-spaces, and that the existence of a free Boolean $F'$-group which is not a $P$-space is equivalent to that of selective ultrafilters on $ω$.
Two (strongly) zero-dimensional Lindelöf topological groups whose product has positive covering dimension are constructed. An example of a Lindelöf (strongly) zero-dimensional space whose free and free Abelian topological groups are not strongly zero-dimensional is given.
It is proved that the square of the Sorgenfrey line has infinite covering dimension.
A group topology is said to be linear if open subgroups form a base of neighborhoods of the identity element. It is proved that the existence of a nondiscrete extremally disconnected group of Ulam nonmeasurable cardinality with linear topology implies that of a nondiscrete extremally disconnected group of cardinality at most $2^ω$ with linear topology.
Given an arbitrary measurable cardinal $κ$, a nondiscrete Hausdorff extremally disconnected topological group of cardinality $κ$ is constructed.
This is the opening article of the abstract book of conference "Set-Theoretic Topology and Topological Algebra" in honor of professor Alexander Arhangelskii on the occasion of his 80th birthday held in 2018 at Moscow State University.
It is proved that any countable topological group in which the filter of neighborhoods of the identity element is not rapid contains a discrete set with precisely one nonisolated point. This gives a negative answer to Protasov's question on the existence in ZFC of a countable nondiscrete group in which all discrete subsets are closed. It is also proved that the existence of a countable nondiscrete extremally disconnected group implies the existence of a rapid ultrafilter and, hence, a countable nondiscrete extremally disconnected group cannot be constructed in ZFC.
Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis is placed on the application of set-theoretic methods to the study of Boolean topological groups.
It is proved that the existence of a countable extremally disconnected Boolean topological group containing a family of open subgroups whose intersection has empty interior implies the existence of a rapid ultrafilter.