For a compact Hausdorff abelian group K and its subgroup H ≤ K, one defines the g-closure g K (H) of H in K as the subgroup consisting of χ ∈ K such that χ(a n) −→ 0 in T = R/Z for every sequence {a n } inˆK (the Pontryagin dual of K) that converges to 0 in the topology that H induces onˆK. We prove that every countable subgroup of a compact Hausdorff group is g-closed, and thus give a positive...

It is well known that the category KHaus of compact Hausdorff spaces is dually equivalent to the category KRFrm of compact regular frames. By de Vries duality, KHaus is also dually equivalent to the category DeV of de Vries algebras, and so DeV is equivalent to KRFrm, where the latter equivalence can be described constructively through Booleanization. Our purpose here is to lift this circle of ...

We discuss scaling limits of large bipartite planar maps. If p ≥ 2 is a fixed integer, we consider, for every integer n ≥ 2, a random planar map Mn which is uniformly distributed over the set of all rooted 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of Mn, equipped with the graph distance rescaled by the factor n−1...

Let X be the Bennett-Lutzer’s space and Y be the space obtained from X by shrinking the set of all rational numbers to a point. In his book, G.Gao claimed that the space Y is compact. In this paper, we prove that Y is neither countably compact nor Lindelöf, which shows that G.Gao’s claim is not true. Moreover, we prove that Y is strongly paracompact. As an application of this result, we obtain ...

We continue to study one of the classic problems in general topology raised by P. S. Alexandrov: when a Hausdorff space X has continuous bijection (a condensation) onto compactum? concentrate on situation not only but also $$X\setminus Y$$ can be condensed compactum whenever cardinality Y does exceed certain $$\tau$$ .

Given $$(X,\omega )$$ compact Kähler manifold and $$\psi \in \mathcal {M}^{+}\subset PSH(X,\omega a model type envelope with non-zero mass, i.e. fixed potential determining singularity such that $$\int _{X}(\omega +dd^{c}\psi )^{n}>0$$ , we prove the -$$ relative finite energy class $$\mathcal {E}^{1}(X,\omega ,\psi becomes complete metric space if endowed distance d which generalizes well-know...