‎A topological space is called submaximal if each of its dense subsets is open and is called nodec if each of its nowhere dense ea subsets is closed‎. ‎Here‎, ‎we study a variety of spaces some of which have already been studied in $C(X)$‎. ‎Among them are‎, ‎most importantly‎, ‎quasi $P$-spaces and pointwise quasi $P$-spaces‎. ‎We obtain some new useful topological characterizations of quasi $...

This paper is based on a series of 4 lectures delivered at Groups St Andrews 2013. The main theme of the lectures was distinguishing finitely generated residually finite groups by their finite quotients. The purpose of this paper is to expand and develop the lectures. The paper is organized as follows. In §2 we collect some questions that motivated the lectures and this article, and in §3 discu...

We provide elementary proofs of the Nielsen-Schreier Theorem and the Kurosh Subgroup Theorem via wreath products. Our proofs are diagrammatic in nature and work simultaneously in the abstract and profinite categories. A new proof that open subgroups of quasifree profinite groups are quasifree is also given.

based on a complete heyting algebra, we modify the definition oflattice-valued fuzzifying convergence space using fuzzy inclusionorder and construct in this way a cartesian-closed category, calledthe category of $l-$ordered fuzzifying convergence spaces, in whichthe category of $l-$fuzzifying topological spaces can be embedded.in addition, two new categories are introduced, which are called the...

A profinite group G is just infinite if every closed normal subgroup of G is of finite index. We prove that an infinite profinite group is just infinite if and only if, for every open subgroup H of G, there are only finitely many open normal subgroups of G not contained in H . This extends a result recently established by Barnea, Gavioli, Jaikin-Zapirain, Monti and Scoppola in [1], who proved t...

Let $p:Xlo B$ be a locally trivial principal G-bundle and $wt{p}:wt{X}lo B$ be a locally trivial principal $wt{G}$-bundle. In this paper, by using the structure of principal bundles according to transition functions, we show that $wt{G}$ is a covering group of $G$ if and only if $wt{X}$ is a covering space of $X$. Then we conclude that a topological space $X$ with non-simply connected universal...

We prove that the normalizer of any diffuse amenable subalgebra of a free group factor L(Fr) generates an amenable von Neumann subalgebra. Moreover, any II1 factor of the form Q ⊗̄L(Fr), with Q an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group Fr, 2 ≤ r ≤ ∞, on a probability ...

We show that the quasiconvex subgroups in doubles of certain negatively curved groups are closed in the profinite topology. This allows us to construct the first known large family of hyperbolic 3-manifolds such that any finitely generated subgroup of the fundamental group of any member of the family is closed in the profinite topology.

In [1,2,3], A. C. Baker and J.W. Baker studied the subspace Ma(S) of the convolution measure algebra M, (S) of a locally compact semigroup. H. Dzinotyiweyi in [5,7] considers an analogous measure space on a large class of C-distinguished topological semigroups containing all completely regular topological semigroups. In this paper, we extend the definitions to study the weighted semigroup ...

In this article we introduce the concept of $z^circ$-filter on a topological space $X$. We study and investigate the behavior of $z^circ$-filters and compare them  with corresponding ideals, namely, $z^circ$-ideals of $C(X)$,  the ring of real-valued continuous functions on a completely regular Hausdorff space $X$. It is observed that $X$ is a compact space if and only if every $z^circ$-filter ...