### A constructive proof of the Tychonoff's theorem for locales

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In this paper, it is proved that a first-countable paratopological group has a regular ${G}_{\delta}$-diagonal, which gives an affirmative answer to Arhangel’skii and Burke’s question [Spaces with a regular ${G}_{\delta}$-diagonal, Topology Appl. 153 (2006), 1917–1929]. If $G$ is a symmetrizable paratopological group, then $G$ is a developable space. We also discuss copies of ${S}_{\omega}$ and of ${S}_{2}$ in paratopological groups and generalize some Nyikos [Metrizability and the Fréchet-Urysohn property in topological groups, Proc. Amer. Math....

Let $T:[0,\infty )\to L\left(E\right)$ be a ${C}_{0}$-semigroup with unbounded generator $A:D\left(A\right)\to E$. We prove that $\left(T\right(t)x-x)/t$ has generically a very irregular behaviour for $x\notin D\left(A\right)$ as $t\to 0+$.

In [5], W. Taylor shows that each particular compact polyhedron can be characterized in the class of all metrizable spaces containing an arc by means of first order properties of its clone of continuous operations. We will show that such a characterization is possible in the class of compact spaces and in the class of Hausdorff spaces containing an arc. Moreover, our characterization uses only the first order properties of the monoid of self-maps. Also, the possibility of characterizing the closed...

This paper discusses properties of the Higson corona by means of a quotient on coarse ultrafilters on a proper metric space. We use this description to show that the corona functor is faithful and reflects isomorphisms.

We introduce a new (extended) quasi-metric on the so-called dual p-complexity space, which is suitable to give a quantitative measure of the improvement in complexity obtained when a complexity function is replaced by a more efficient complexity function on all inputs, and show that this distance function has the advantage of possessing rich topological and quasi-metric properties. In particular, its induced topology is Hausdorff and completely regular. Our approach is applied to the measurement...

In 1985, V. G. Pestov described a neighborhood base at the identity of free topological groups on a Tychonoff space in terms of the elements of the fine uniformity on the Tychonoff space. In this paper, we extend Postev’s description to the free paratopological groups where we introduce a neighborhood base at the identity of free paratopological groups on any topological space in terms of the elements of the fine quasiuniformity on the space.