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Cantor-connectedness revisited

Robert Lowen (1992)

Commentationes Mathematicae Universitatis Carolinae

Following Preuss' general connectedness theory in topological categories, a connectedness concept for approach spaces is introduced, which unifies topological connectedness in the setting of topological spaces, and Cantor-connectedness in the setting of metric spaces.

Characterizing chainable, tree-like, and circle-like continua

Taras Banakh, Zdzisław Kosztołowicz, Sławomir Turek (2011)

Colloquium Mathematicae

We prove that a continuum X is tree-like (resp. circle-like, chainable) if and only if for each open cover 𝓤₄ = {U₁,U₂,U₃,U₄} of X there is a 𝓤₄-map f: X → Y onto a tree (resp. onto the circle, onto the interval). A continuum X is an acyclic curve if and only if for each open cover 𝓤₃ = {U₁,U₂,U₃} of X there is a 𝓤₃-map f: X → Y onto a tree (or the interval [0,1]).

Connected economically metrizable spaces

Taras Banakh, Myroslava Vovk, Michał Ryszard Wójcik (2011)

Fundamenta Mathematicae

A topological space is non-separably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected sequential topological space X is the image of a non-separably connected complete metric space X under a monotone quotient map. The metric d X of the space X is economical in the sense that for each infinite subspace A ⊂ X the cardinality of the set d X ( a , b ) : a , b A does not exceed the density of A, | d X ( A × A ) | d e n s ( A ) . The construction of the space X determines a functor : Top...

Connected Hausdorff subtopologies

Jack R. Porter (2001)

Commentationes Mathematicae Universitatis Carolinae

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 π -weight and no connected, Hausdorff subtopology is provided.

Connectedness and local connectedness of topological groups and extensions

Ofelia Teresa Alas, Mihail G. Tkachenko, Vladimir Vladimirovich Tkachuk, Richard Gordon Wilson (1999)

Commentationes Mathematicae Universitatis Carolinae

It is shown that both the free topological group F ( X ) and the free Abelian topological group A ( X ) on a connected locally connected space X are locally connected. For the Graev’s modification of the groups F ( X ) and A ( X ) , the corresponding result is more symmetric: the groups F Γ ( X ) and A Γ ( X ) are connected and locally connected if X is. However, the free (Abelian) totally bounded group F T B ( X ) (resp., A T B ( X ) ) is not locally connected no matter how “good” a space X is. The above results imply that every non-trivial continuous homomorphism...

Connections between connected topological spaces on the set of positive integers

Paulina Szczuka (2013)

Open Mathematics

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...

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