We prove that a semiregular topological space X is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups.1991 Mathematics Subject Classification. 54D10; 54D15; 54E15; 22A30.
An H-closed quasitopological group is a Hausdorff quasitopological group which is contained in each Hausdorff quasitopological group as a closed subspace. We obtained a sufficient condition for a quasitopological group to be H-closed, which allowed us to solve a problem by Arhangel'skii and Choban and to show that a topological group G is H-closed in the class of quasitopological groups if and only if G is Raǐkov-complete. Also we present examples of non-compact quasitopological groups whose topological spaces are H-closed.
For any topological space $X$ we study the relation between the universal
uniformity $\mathcal U_X$, the universal quasi-uniformity $q\mathcal U_X$ and
the universal pre-uniformity $p\mathcal U_X$ on $X$. For a pre-uniformity
$\mathcal U$ on a set $X$ and a word $v$ in the two-letter alphabet $\{+,-\}$
we define the verbal power $\mathcal U^v$ of $\mathcal U$ and study its
boundedness numbers $\ell(\mathcal U^v)$ and $\bar \ell(\mathcal U^v)$. The
boundedness numbers of the (Boolean operations over) the verbal powers of the
canonical pre-uniformities $p\mathcal U_X$, $q\mathcal U_X$ and $\mathcal U_X$
yield new cardinal characteristics $\ell^v(X)$, $\bar \ell^v(X)$, $q\ell^v(X)$,
$q\bar \ell^v(X)$, $u\ell(X)$ of a topological space $X$, which generalize all
known cardinal topological invariants related to (star)-covering properties. We
study the relation of the new cardinal invariants $\ell^v$, $\bar \ell^v$ to
classical cardinal topological invariants such as Lindel\"of number $\ell$,
density $d$, and spread $s$. The simplest new verbal cardinal invariant is the
foredensity $\ell^-(X)$ defined for a topological space $X$ as the smallest
cardinal $\kappa$ such that for any neighborhood assignment $(O_x)_{x\in X}$
there is a subset $A\subset X$ of cardinality $|A|\le\kappa$ that meets each
neighborhood $O_x$, $x\in X$. It is clear that $\ell^-(X)\le d(X)\le
\ell^-(X)\cdot \chi(X)$. We shall prove that $\ell^-(X)=d(X)$ if
$|X|<\aleph_\omega$. On the other hand, for every singular cardinal $\kappa$
(with $\kappa\le 2^{2^{cf(\kappa)}}$) we construct a (totally disconnected)
$T_1$-space $X$ such that $\ell^-(X)=cf(\kappa)<\kappa=|X|=d(X)$.Comment: 20 pages, many diagram
A topologized semilattice X is complete if each non-empty chain C ⊂ X has inf C ∈C and sup C ∈C. It is proved that for any complete subsemilattice X of a functionally Hausdorff semitopological semilattice Y the partial order P = {(x, y) ∈ X × X : xy = x} of X is closed in Y × Y and hence X is closed in Y . This implies that for any continuous homomorphism h : X → Y from a compete topologized semilattice X to a functionally Hausdorff semitopological semilattice Y the image h(X) is closed in Y . The functional Hausdorffness of Y in these two results can be replaced by the weaker separation axiom T 2δ , defined in this paper.
We construct a metrizable Lawson semitopological semilattice X whose partial order ≤ X = {(x, y) ∈ X × X : xy = x} is not closed in X × X. This resolves a problem posed earlier by the authors.1991 Mathematics Subject Classification. 54A20, 06A12, 22A26, 37B05.
Given a Tychonoff space X, let F (X) and A(X) be respectively the free topological group and the free Abelian topological group over X in the sense of Markov. In this paper, we consider two topological properties of F (X) or A(X), namely the countable tightness and G-base. We provide some characterizations of the countable tightness and G-base of F (X) and A(X) for various special classes of spaces X. Furthermore, we also study the countable tightness and G-base of some Fn(X) of F (X).2000 Mathematics Subject Classification. Primary 54H11, 22A05; Secondary 54E20; 54E35; 54D50; 54D55.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.