A non-connected, Hausdorff space with a countable network has a connected Hausdorff-subtopology iff the space is not-H-closed. This result answers two questions posed by Tkačenko, Tkachuk, Uspenskij, and Wilson [Comment. Math. Univ. Carolinae 37 (1996), 825–841]. A non-H-closed, Hausdorff space with countable -weight and no connected, Hausdorff subtopology is provided.
We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if is a connected locally compact Abelian subgroup of a Hausdorff topological group and the quotient space is sequentially connected, then so is .
It is shown that both the free topological group and the free Abelian topological group on a connected locally connected space are locally connected. For the Graev’s modification of the groups and , the corresponding result is more symmetric: the groups and are connected and locally connected if is. However, the free (Abelian) totally bounded group (resp., ) is not locally connected no matter how “good” a space is. The above results imply that every non-trivial continuous homomorphism...
In this article we define the -topology on some rings of quotients of . Using this, we equip the classical ring of quotients of with the -topology and we show that with the -topology is in fact a subspace of with the -topology. Characterization of the components of rings of quotients of is given and using this, it turns out that with the -topology is connected if and only if is a pseudocompact almost -space, if and only if with -topology is connected. We also observe that...
We discuss the ideas of Morse decompositions and index filtrations for isolated invariant sets for both single-valued and multi-valued maps. We introduce the definition of connection matrix pairs and present the theorem of their existence. Connection matrix pair theory for multi-valued maps is used to show that connection matrix pairs obey the continuation property. We conclude by addressing applications to numerical analysis. This paper is primarily an overview of the papers [R1] and [R2].
In this paper we introduce a connected topology T on the set ℕ of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ℕ which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of...
We characterize those Baire one functions f for which the diagonal product x → (f(x), g(x)) has a connected graph whenever g is approximately continuous or is a derivative.