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Answering a question of Garbuli\'nska-W\c{e}grzyn and Kubi\'s, we prove that Gurarii operators form a dense $G_\delta$-set in the space $\mathcal B(\mathbb G)$ of all nonexpansive operators on the Gurarii space $\mathbb G$, endowed with the strong operator topology. This implies that the set of universal operators on $\mathbb G$ form a residual set in $\mathcal B(\mathbb G)$., Comment: 12 pages

Let $\mathcal C$ be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup $X$ is called $\mathcal C$-$closed$ if $X$ is closed in each topological semigroup $Y\in \mathcal C$ containing $X$ as a discrete subsemigroup; $X$ is $projectively$ $\mathcal C$-$closed$ if for each congruence $\approx$ on $X$ the quotient semigroup $X/_\approx$ is $\mathcal C$-closed. A semigroup $X$ is called $chain$-$finite$ if for any infinite set $I\subseteq X$ there are elements $x,y\in I$ such that $xy\notin\{x,y\}$. We prove that a semigroup $X$ is $\mathcal C$-closed if it admits a homomorphism $h:X\to E$ to a chain-finite semilattice $E$ such that for every $e\in E$ the semigroup $h^{-1}(e)$ is $\mathcal C$-closed. Applying this theorem, we prove that a commutative semigroup $X$ is $\mathcal C$-closed if and only if $X$ is periodic, chain-finite, all subgroups of $X$ are bounded, and for any infinite set $A\subseteq X$ the product $AA$ is not a singleton. A commutative semigroup $X$ is projectively $\mathcal C$-closed if and only if $X$ is chain-finite, all subgroups of $X$ are bounded and the union $H(X)$ of all subgroups in $X$ has finite complement $X\setminus H(X)$., Comment: 19 pages

Let L(X) be the free locally convex space over a Tychonoff space X. We prove that the following assertions are equivalent: (i) every functionally bounded subset of X is finite, (ii) L(X) is semi-reflexive, (iii) L(X) has the Grothendieck property, (iv) L(X) is semi-Montel. We characterize those spaces X, for which L(X) is c0-quasibarrelled, distinguished or a (d f)-space. If X is a convergent sequence, then L(X) has the Glicksberg property, but the space L(X) endowed with its Mackey topology does not have the Schur property.

For any metric space $X$ endowed with the action of a group $G$, and two $n$-gons $\vec x=(x_1,\dots,x_n)\in X^n$ and $\vec y=(y_1,\dots,y_n)\in X^n$ in $X$, we introduce the $G$-deviation $d(G\vec x,\vec y\,)$ of $\vec x$ from $\vec y$ as the distance in $X^n$ from $\vec y$ to the $G$-orbit $G\vec x$ of $\vec x$ in the $n$-th power $X^n$ of $X$. For some groups $G$ of affine transformations of the complex plane, we deduce simple-to-apply formulas for calculating the $G$-derviation between $n$-gons on the complex plane. We apply these formulas for defining new measures of asymmetry of triangles. These new measures can be applied in Electric Power Engineering for evaluating the quality of 3-phase electric power. One of such measures, namely the affine deviation, is espressible via the unbalance degree, which is a standard characteristic of quality of three-phase electric power., 14 pages

A subset $B$ of a group $G$ is called a basis of $G$ if each element $g\in G$ can be written as $g=ab$ for some elements $a,b\in B$. The smallest cardinality $|B|$ of a basis $B\subseteq G$ is called the basis size of $G$ and is denoted by $r[G]$. We prove that each finite group $G$ has $r[G]>\sqrt{|G|}$. If $G$ is Abelian, then $r[G]\ge \sqrt{2|G|-|G|/|G_2|}$, where $G_2=\{g\in G:g^{-1} = g\}$. Also we calculate the basis sizes of all Abelian groups of order $\le 60$ and all non-Abelian groups of order $\le 40$.

A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-dimensional compact subsets. It turns out that $$\varphi $$ φ is a unique infinite-dimensional locally convex space which is a $$k_{\mathbb {R}}$$ k R -space containing no infinite-dimensional compact subsets. Applications to spaces $$C_{p}(X)$$ C p ( X ) are provided.

A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid S is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1.

The path component space of a topological space $X$ is the quotient space $\pi_0(X)$ whose points are the path components of $X$. We show that every Tychonoff space $X$ is the path-component space of a Tychonoff space $Y$ of weight $w(Y)=w(X)$ such that the natural quotient map $Y\to \pi_0(Y)=X$ is a perfect map. Hence, many topological properties of $X$ transfer to $Y$. We apply this result to construct a compact space $X\subset \mathbb{R}^3$ for which the fundamental group $\pi_1(X,x_0)$ is an uncountable, cosmic, $k_{\omega}$-topological group but for which the canonical homomorphism $\psi:\pi_1(X,x_0)\to \check{\pi}_1(X,x_0)$ to the first shape homotopy group is trivial., Comment: 12 pages

Let$${\mathcal {B}}$$Bbe a class of finite-dimensional Banach spaces. A$${\mathcal {B}}$$B-decomposed Banach spaceis a Banach spaceXendowed with a family$${\mathcal {B}}_X\subset {\mathcal {B}}$$BX⊂Bof subspaces ofXsuch that each$$x\in X$$x∈Xcan be uniquely written as the sum of an unconditionally convergent series$$\sum _{B\in {\mathcal {B}}_X}x_B$$∑B∈BXxBfor some$$(x_B)_{B\in {\mathcal {B}}_X}\in \prod _{B\in {\mathcal {B}}_X}B$$(xB)B∈BX∈∏B∈BXB. For every$$B\in {\mathcal {B}}_X$$B∈BXlet$$\mathrm {pr}_B:X\rightarrow B$$prB:X→Bdenote the coordinate projection. Let$$C\subset [-1,1]$$C⊂[-1,1]be a closed convex set with$$C\cdot C\subset C$$C·C⊂C. TheC-decomposition constant$$K_C$$KCof a$${\mathcal {B}}$$B-decomposed Banach space$$(X,{\mathcal {B}}_X)$$(X,BX)is the smallest number$$K_C$$KCsuch that for every function$$\alpha :{\mathcal {F}}\rightarrow C$$α:F→Cfrom a finite subset$${\mathcal {F}}\subset {\mathcal {B}}_X$$F⊂BXthe operator$$T_\alpha =\sum _{B\in {\mathcal {F}}}\alpha (B)\cdot \mathrm {pr}_B$$Tα=∑B∈Fα(B)·prBhas norm$$\Vert T_\alpha \Vert \le K_C$$‖Tα‖≤KC. By$$\varvec{{\mathcal {B}}}_C$$BCwe denote the class of$${\mathcal {B}}$$B-decomposed Banach spaces withC-decomposition constant$$K_C\le 1$$KC≤1. Using the technique of Fraïssé theory, we construct a rational$${\mathcal {B}}$$B-decomposed Banach space$$\mathbb {U}_C\in \varvec{{\mathcal {B}}}_C$$UC∈BCwhich contains an almost isometric copy of each$${\mathcal {B}}$$B-decomposed Banach space$$X\in \varvec{{\mathcal {B}}}_C$$X∈BC. If$${\mathcal {B}}$$Bis the class of all 1-dimensional (resp. finite-dimensional) Banach spaces, then$$\mathbb {U}_{C}$$UCis isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional (f.d.) basis, constructed by Pełczyński (and Wojtaszczyk).

For a hypergraph H = ( V , E ) , a subfamily C ⊆ E is called a cover of the hypergraph if ⋃ C = ⋃ E . A cover C is called minimal if each cover D ⊆ C of the hypergraph H coincides with C . We prove that for a hypergraph H the following conditions are equivalent: (i) each countable subhypergraph of H has a minimal cover; (ii) each non-empty subhypergraph of H has a maximal edge; (iii) H contains no isomorphic copy of the hypergraph ( ω , ω ) . This characterization implies that a countable hypergraph ( V , E ) has a minimal cover if every infinite set I ⊆ V contains a finite subset F ⊆ I such that the family of edges E F ≔ { E ∈ E : F ⊆ E } is finite. Also we prove that a hypergraph ( V , E ) has a minimal cover if sup { | E | : E ∈ E } ω or for every v ∈ V the family E v ≔ { E ∈ E : v ∈ E } is finite.

In this paper we establish a connection between categorical closedness and topologizability of semigroups. In particular, for a class $\mathsf T_{\!1}\mathsf S$ of $T_1$ topological semigroups we prove that a countable semigroup $X$ with finite-to-one shifts is injectively $\mathsf T_{\!1}\mathsf S$-closed if and only if $X$ is $\mathsf{T_{\!1}S}$-nontopologizable in the sense that every $T_1$ semigroup topology on $X$ is discrete. Moreover, a countable cancellative semigroup $X$ is absolutely $\mathsf T_{\!1}\mathsf S$-closed if and only if every homomorphic image of $X$ is $\mathsf T_{\!1}\mathsf S$-nontopologizable. Also, we introduce and investigate a notion of a polybounded semigroup. It is proved that a countable semigroup $X$ with finite-to-one shifts is polybounded if and only if $X$ is $\mathsf T_{\!1}\mathsf S$-closed if and only if $X$ is $\mathsf T_{\!z}\mathsf S$-closed, where $\mathsf T_{\!z}\mathsf S$ is a class of zero-dimensional Tychonoff topological semigroups. We show that polyboundedness provides an automatic continuity of the inversion in $T_1$ paratopological groups and prove that every cancellative polybounded semigroup is a group., 25 pages

A topologized semilattice $X$ is called complete if each non-empty chain $C\subset X$ has $\inf C\in\bar C$ and $\sup C\in\bar C$. We prove that for any continuous homomorphism $h:X\to Y$ from a complete topologized semilattice $X$ to a sequential Hausdorff semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$., 6 pages

We find (completeness type) conditions on topological semilattices $X,Y$ guaranteeing that each continuous homomorphism $h:X\to Y$ has closed image $h(X)$ in $Y$., Comment: 12 pages

The famous Rosenthal-Lacey theorem asserts that for each infinite compact space $K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c_{0}$ or $\ell_{2}$. The aim of the paper is to study a natural variant of this result for the space $C_{p}(K)$ of continuous real-valued maps on $K$ with the pointwise topology. Following famous Josefson-Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson-Nissenzweig property, briefly, the JNP) for $C_{p}$-spaces. We prove: For a Tychonoff space $X$ the space $C_p(X)$ satisfies the JNP if and only if $C_p(X)$ has a quotient isomorphic to $c_{0}$ (with the product topology of $\mathbb R^\mathbb{N}$) if and only if $C_{p}(X)$ contains a complemented subspace, isomorphic to $c_0$. For a pseudocompact space $X$ the space $C_p(X)$ has the JNP if and only if $C_p(X)$ has a complemented metrizable infinite-dimensional subspace. This applies to show that for a Tychonoff space $X$ the space $C_p(X)$ has a complemented subspace isomorphic to $\mathbb R^{\mathbb N}$ or $c_0$ if and only if $X$ is not pseudocompact or $C_p(X)$ has the JNP. The space $C_{p}(\beta\mathbb{N})$ contains a subspace isomorphic to $c_0$ and admits a quotient isomorphic to $\ell_{\infty}$ but fails to have a quotient isomorphic to $c_{0}$. An example of a compact space $K$ without infinite convergent sequences with $C_{p}(K)$ containing a complemented subspace isomorphic to $c_{0}$ is constructed., Comment: 14 pages

In the paper we introduce and study a new family of “small” sets which is tightly connected with two well known σ-ideals: of Haar-null sets and of Haar-meager sets. We define a subset A of a topological group X to be null-finite if there exists a convergent sequence (xn)n∈ω in X such that for every x ∈ X the set {n ∈ ω : x + xn ∈ A} is finite. We prove that each null-finite Borel set in a complete metric Abelian group is Haar-null and Haar-meager. The Borel restriction in the above result is essential as each non-discrete metric Abelian group is the union of two null-finite sets. Applying null-finite sets to the theory of functional equations and inequalities, we prove that a mid-point convex function f : G → ℝ defined on an open convex subset G of a metric linear space X is continuous if it is upper bounded on a subset B which is not null-finite and whose closure is contained in G. This gives an alternative short proof of a known generalization of the Bernstein–Doetsch theorem (saying that a mid-point convex function f: G → ℝ defined on an open convex subset G of a metric linear space X is continuous if it is upper bounded on a non-empty open subset B of G). Since Borel Haar-finite sets are Haar-meager and Haar-null, we conclude that a mid-point convex function f: G → ℝ defined on an open convex subset G of a complete linear metric space X is continuous if it is upper bounded on a Borel subset B ⊂ G which is not Haar-null or not Haar-meager in X. The last result resolves an old problem in the theory of functional equations and inequalities posed by Baron and Ger in 1983.

The Golomb space $\mathbb N_\tau$ is the set $\mathbb N$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+bn\}_{n=0}^\infty$ with coprime $a,b$. We prove that the Golomb space $\mathbb N_\tau$ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $\Pi$ of prime numbers is a dense metrizable subspace of $\mathbb N_\tau$, and each homeomorphism $h$ of $\mathbb N_\tau$ has the following properties: $h(1)=1$, $h(\Pi)=\Pi$ and $\Pi_{h(x)}=h(\Pi_x)$ for all $x\in\mathbb N$. Here by $\Pi_x$ we denote the set of prime divisors of $x$., Comment: 12 pages

Given a partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_\alpha)_{\alpha\in P}$ of subsets of $X\times X$ such that $U_\beta\subset U_\alpha$ for all $\alpha\le\beta$ in $P$ and for every $x\in X$ the family $(U_\alpha[x])_{\alpha\in P}$ of balls $U_\alpha[x]=\{y\in X:(x,y)\in U_\alpha\}$ is a neighborhood base at $x$. A $P$-base $(U_\alpha)_{\alpha\in P}$ for $X$ is called locally uniform if the family of entourages $(U_\alpha U_\alpha^{-1}U_\alpha)_{\alpha\in P}$ remains a $P$-base for $X$. A topological space is first-countable if and only if it has an $\omega$-base. By Moore's Metrization Theorem, a topological space is metrizable if and only if it is a $T_0$-space with a locally uniform $\omega$-base. In the paper we shall study topological spaces possessing a (locally uniform) $\omega^\omega$-base. Our results show that spaces with an $\omega^\omega$-base share some common properties with first countable spaces, in particular, many known upper bounds on the cardinality of first-countable spaces remain true for countably tight $\omega^\omega$-based topological spaces. On the other hand, topological spaces with a locally uniform $\omega^\omega$-base have many properties, typical for generalized metric spaces. Also we study Tychonoff spaces whose universal (pre- or quasi-) uniformity has an $\omega^\omega$-base and show that such spaces are close to being $\sigma$-compact., Comment: 105 pages

The famous Rosenthal-Lacey theorem asserts that for each infinite compact set $K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c$ or $\ell_{2}$. What is the case when the uniform topology of $C(K)$ is replaced by the pointwise topology? Is it true that $C_p(X)$ always has an infinite-dimensional separable (or better metrizable) quotient? In this paper we prove that for a Tychonoff space $X$ the function space $C_p(X)$ has an infinite-dimensional metrizable quotient if $X$ either contains an infinite discrete $C^*$-embedded subspace or else $X$ has a sequence $(K_n)_{n\in\mathbb N}$ of compact subsets such that for every $n$ the space $K_n$ contains two disjoint topological copies of $K_{n+1}$. Applying the latter result, we show that under $\lozenge$ there exists a zero-dimensional Efimov space $K$ whose function space $C_{p}(K)$ has an infinite-dimensional metrizable quotient. These two theorems essentially improve earlier results of K\k{a}kol and \'Sliwa on infinite-dimensional separable quotients of $C_p$-spaces., Comment: 8 pages

We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear isometry of the Banach spaces. This resolves the famous Tingley's problem in the class of 2-dimensional Banach spaces., Comment: 8 pages

A Banach space $X$ has the $Mazur$-$Ulam$ $property$ if any isometry from the unit sphere of $X$ onto the unit sphere of any other Banach space $Y$ extends to a linear isometry of the Banach spaces $X,Y$. A Banach space $X$ is called $smooth$ if the unit ball has a unique supporting functional at each point of the unit sphere. We prove that each non-smooth 2-dimensional Banach space has the Mazur-Ulam property., Comment: 13 pages

A Hausdorff topological space X is called superconnected (resp. coregular) if for any nonempty open sets U 1 , … U n ⊆ X , the intersection of their closures U ‾ 1 ∩ … ∩ U ‾ n is not empty (resp. the complement X ∖ ( U ‾ 1 ∩ … ∩ U ‾ n ) is a regular topological space). A canonical example of a coregular superconnected space is the projective space Q P ∞ of the topological vector space Q ω = { ( x n ) n ∈ ω ∈ Q ω : | { n ∈ ω : x n ≠ 0 } | ω } over the field of rationals Q . The space Q P ∞ is the quotient space of Q ω ∖ { 0 } ω by the equivalence relation x ∼ y iff Q ⋅ x = Q ⋅ y . We prove that every countable second-countable coregular space is homeomorphic to a subspace of Q P ∞ , and a topological space X is homeomorphic to Q P ∞ if and only if X is countable, second-countable, and admits a decreasing sequence of closed sets ( X n ) n ∈ ω such that (i) X 0 = X , ⋂ n ∈ ω X n = ∅ , (ii) for every n ∈ ω and a nonempty relatively open set U ⊆ X n the closure U ‾ contains some set X m , and (iii) for every n ∈ ω the complement X ∖ X n is a regular topological space. Using this topological characterization of Q P ∞ we find topological copies of the space Q P ∞ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.

The $Golomb$ $space$ (resp. the $Kirch$ $space$) is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic progressions $a+b\mathbb N_0=\{a+bn:n\ge 0\}$ where $a\in\mathbb N$ and $b$ is a (square-free) number, coprime with $a$. It is known that the Golomb space (resp. the Kirch space) is connected (and locally connected). By a recent result of Banakh, Spirito and Turek, the Golomb space has trivial homeomorphism group and hence is topologically rigid. In this paper we prove the topological rigidity of the Kirch space., 12 pages

For a metric space $X$, let $\mathsf FX$ be the space of all nonempty finite subsets of $X$ endowed with the largest metric $d^1_{\mathsf FX}$ such that for every $n\in\mathbb N$ the map $X^n\to\mathsf FX$, $(x_1,\dots,x_n)\mapsto \{x_1,\dots,x_n\}$, is non-expanding with respect to the $\ell^1$-metric on $X^n$. We study the completion of the metric space $\mathsf F^1\!X=(\mathsf FX,d^1_{\mathsf FX})$ and prove that it coincides with the space $\mathsf Z^1\!X$ of nonempty compact subsets of $X$ that have zero length (defined with the help of graphs). We prove that each subset of zero length in a metric space has 1-dimensional Hausdorff measure zero. A subset $A$ of the real line has zero length if and only if its closure is compact and has Lebesgue measure zero. On the other hand, for every $n\ge 2$ the Euclidean space $\mathbb R^n$ contains a compact subset of 1-dimensional Hausdorff measure zero that fails to have zero length., 12 pages

A topological group $X$ is called $duoseparable$ if there exists a countable set $S\subseteq X$ such that $SUS=X$ for any neighborhood $U\subseteq X$ of the unit. We construct a functor $F$ assigning to each (abelian) topological group $X$ a duoseparable (abelain-by-cyclic) topological group $FX$, containing an isomorphic copy of $X$. In fact, the functor $F$ is defined on the category of unital topologized magmas. Also we prove that each $\sigma$-compact locally compact abelian topological group embeds into a duoseparable locally compact abelian-by-countable topological group., Comment: 9 pages

For any compact Hausdorff space $K$ we construct a canonical finitary coarse structure $\mathcal E_{X,K}$ on the set $X$ of isolated points of $K$. This construction has two properties: $\bullet$ If a finitary coarse space $(X,\mathcal E)$ is metrizable, then its coarse structure $\mathcal E$ coincides with the coarse structure $\mathcal E_{X,\bar X}$ generated by the Higson compactification $\bar X$ of $X$; $\bullet$ A compact Hausdorff space $K$ coincides with the Higson compactification of the coarse space $(X,\mathcal E_{X,K})$ if the set $X$ is dense in $K$ and the space $K$ is Frechet-Urysohn. This implies that a compact Hausdorff space $K$ is homeomorphic to the Higson corona of some finitary coarse space if one of the following conditions holds: (i) $K$ is perfectly normal; (ii) $K$ has weight $w(K)\le\omega_1$ and character $\chi(K), Comment: 20 pages

We discuss various modifications of separability, precompactnmess and narrowness in topological groups and test those modifications in the permutation groups $S(X)$ and $S_{, Comment: 6 pages

We prove that any isometry between the unit spheres of $C^2$-smooth (more generally, absolutely smooth) smooth Banach spaces extends to a linear isometry of the Banach spaces. This answers the famous Tingley's problem in the class of absolutely smooth $2$-dimensional Banach spaces., 22 pages

For a topological space $X$ its reflection in a class $\mathsf T$ of topological spaces is a pair $(\mathsf T X,i_X)$ consisting of a space $\mathsf T X\in\mathsf T$ and continuous map $i_X:X\to \mathsf T X$ such that for any continuous map $f:X\to Y$ to a space $Y\in\mathsf T$ there exists a unique continuous map $\bar f:\mathsf T X\to Y$ such that $f=\bar f\circ i_X$. In this paper for an infinite cardinal $\kappa$ and a nonempty set $M$ of ultrafilters on $\kappa$, we study the reflections of topological spaces in the classes $\mathsf H_\kappa$ of $\kappa$-bounded Hausdorff spaces and $\mathsf H_M$ of $M$-compact Hausdorff spaces (a topological space $X$ is $\kappa$-bounded if the closures of subsets of cardinality $\le\kappa$ in $X$ are compact; $X$ is $M$-compact if any function $x:\kappa\to X$ has a $p$-limit in $M$ for every ultrafilter $p\in M$)., Comment: 17 pages

Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set $X$ endowed with $n$ pairwise comparable topologies $\tau_1\subset\dots\subset\tau_n$, by repeated application of the operations of complement and closure in the topologies $\tau_1,\dots,\tau_n$ to a subset $A\subset X$ we can obtains at most $2K(n)=2\sum_{i,j=0}^n\binom{i+j}{i}\binom{i+j}{j}$ distinct sets., Comment: 20 pages

We study the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse, etc.) determined by the sizes of their elements. The obtained results are applied to the Cech–Stone compactification 𝛽G of the group G. In particular, it is shown that the closure of the minimal ideal 𝛽G has the F𝜎𝛿 type.

Given a $T_0$ paratopological group $G$ and a class $\mathcal C$ of continuous homomorphisms of paratopological groups, we define the $\mathcal C$-$semicompletion$ $\mathcal C[G)$ and $\mathcal C$-$completion$ $\mathcal C[G]$ of the group $G$ that contain $G$ as a dense subgroup, satisfy the $T_0$-separation axiom and have certain universality properties. For special classes $\mathcal C$, we present some necessary and sufficient conditions on $G$ in order that the (semi)completions $\mathcal C[G)$ and $\mathcal C[G]$ be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group $G$ whose $\mathcal C$-semicompletion $\mathcal C[G)$ fails to be a $T_1$-space, where $\mathcal C$ is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group $G$ contains an $\omega$-bounded sequentially compact subgroup $H$ such that $H$ is a topological group but its closure in $G$ fails to be a subgroup., Comment: 10 pages

Let ( X , d ) be a complete, bounded, metric space. For a nonempty, closed subset A of X denote by C ⁎ ( A × A ) the set of all continuous, bounded, real-valued functions on A × A . Denote by C † = ⋃ { C ⁎ ( A × A ) | A is a nonempty closed subset of X } the set of all partial, continuous and bounded functions. We prove that there exists a linear, regular extension operator from C † endowed with the topology of convergence in the Hausdorff distance of graphs of partial functions to the space C ⁎ ( X × X ) with the topology of uniform convergence on compact sets. The constructed extension operator preserves constant functions, pseudometrics, metrics and admissible metrics. For a fixed, nonempty, closed subset A of X the restricted extension operator from C ⁎ ( A × A ) to C ⁎ ( X × X ) is continuous with respect to the topologies of pointwise convergence, uniform convergence on compact sets and uniform convergence considered on both C ⁎ ( A × A ) and C ⁎ ( X × X ) .

A topological space $X$ has the strong Pytkeev property at a point $x\in X$ if there exists a countable family $\mathcal N$ of subsets of $X$ such that for each neighborhood $O_x\subset X$ and subset $A\subset X$ accumulating at $x$, there is a set $N\in\mathcal N$ such that $N\subset O_x$ and $N\cap A$ is infinite. We prove that for any $\aleph_0$-space $X$ and any space $Y$ with the strong Pytkeev property at a point $y\in Y$ the function space $C_k(X,Y)$ has the strong Pytkeev property at the constant function $X\to \{y\}\subset Y$. If the space $Y$ is rectifiable, then the function space $C_k(X,Y)$ is rectifiable and has the strong Pytkeev property at each point. We also prove that for any pointed spaces $(X_n,*_n)$, $n\in\omega$, with the strong Pytkeev property their Tychonoff product and their small box-product both have the strong Pytkeev property at the distinguished point. We prove that a sequential rectifiable space $X$ has the strong Pytkeev property if and only if $X$ is metrizable or contains a clopen submetrizable $k_\omega$-subspace. A locally precompact topological group is metrizable if and only if it contains a dense subgroup with the strong Pytkeev property., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1311.1468

For a Hausdorff topologized semilattice $X$ its $Lawson\;\; number$ $\bar\Lambda(X)$ is the smallest cardinal $\kappa$ such that for any distinct points $x,y\in X$ there exists a family $\mathcal U$ of closed neighborhoods of $x$ in $X$ such that $|\mathcal U|\le\kappa$ and $\bigcap\mathcal U$ is a subsemilattice of $X$ that does not contain $y$. It follows that $\bar\Lambda(X)\le\bar\psi(X)$, where $\bar\psi(X)$ is the smallest cardinal $\kappa$ such that for any point $x\in X$ there exists a family $\mathcal U$ of closed neighborhoods of $x$ in $X$ such that $|\mathcal U|\le\kappa$ and $\bigcap\mathcal U=\{x\}$. We prove that a compact Hausdorff semitopological semilattice $X$ is Lawson (i.e., has a base of the topology consisting of subsemilattices) if and only if $\bar\Lambda(X)=1$. Each Hausdorff topological semilattice $X$ has Lawson number $\bar\Lambda(X)\le\omega$. On the other hand, for any infinite cardinal $\lambda$ we construct a Hausdorff zero-dimensional semitopological semilattice $X$ such that $|X|=\lambda$ and $\bar\Lambda(X)=\bar\psi(X)=cf(\lambda)$. A topologized semilattice $X$ is called (i) $\omega$-$Lawson$ if $\bar\Lambda(X)\le\omega$; (ii) $complete$ if each non-empty chain $C\subset X$ has $\inf C\in\overline{C}$ and $\sup C\in\overline{C}$. We prove that for any complete subsemilattice $X$ of an $\omega$-Lawson semitopological semilattice $Y$, the partial order $\le_X=\{(x,y)\in X\times X:xy=x\}$ of $X$ is closed in $Y\times Y$ and hence $X$ is closed in $Y$. This implies that for any continuous homomorphism $h:X\to Y$ from a compete topologized semilattice $X$ to an $\omega$-Lawson semitopological semilattice $Y$ the image $h(X)$ is closed in $Y$., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1806.02868