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A rigid space admitting compact operators

Paul Sisson (1995)

Studia Mathematica

A rigid space is a topological vector space whose endomorphisms are all simply scalar multiples of the identity map. The first complete rigid space was published in 1981 in [2]. Clearly a rigid space is a trivial-dual space, and admits no compact endomorphisms. In this paper a modification of the original construction results in a rigid space which is, however, the domain space of a compact operator, answering a question that was first raised soon after the existence of complete rigid spaces was...

A solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita (2005)

Fundamenta Mathematicae

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number α ≤ 2 , a topological group G such that G γ is countably compact for all cardinals γ < α, but G α is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under M A c o u n t a b l e . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from M A c o u n t a b l e . However, the question has remained...

A tree π -base for ℝ * without cofinal branches

Fernando Hernández-Hernández (2005)

Commentationes Mathematicae Universitatis Carolinae

We prove an analogue to Dordal’s result in P.L. Dordal, A model in which the base-matrix tree cannot have cofinal branches, J. Symbolic Logic 52 (1980), 651–664. He obtained a model of ZFC in which there is a tree π -base for ℕ * with no ω 2 branches yet of height ω 2 . We establish that this is also possible for ℝ * using a natural modification of Mathias forcing.

Addition theorems, D -spaces and dually discrete spaces

Ofelia Teresa Alas, Vladimir Vladimirovich Tkachuk, Richard Gordon Wilson (2009)

Commentationes Mathematicae Universitatis Carolinae

A neighbourhood assignment in a space X is a family 𝒪 = { O x : x ∈ X } of open subsets of X such that x ∈ O x for any x ∈ X . A set Y ⊆ X is a kernel of 𝒪 if 𝒪 ( Y ) = ⋃ { O x : x ∈ Y } = X . If every neighbourhood assignment in X has a closed and discrete (respectively, discrete) kernel, then X is said to be a D -space (respectively a dually discrete space). In this paper we show among other things that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf P -space is a D -space and we prove an addition...

Algebras and spaces of dense constancies

Angelo Bella, Jorge Martinez, Scott D. Woodward (2001)

Czechoslovak Mathematical Journal

A DC-space (or space of dense constancies) is a Tychonoff space X such that for each f ∈ C ( X ) there is a family of open sets { U i i ∈ I } , the union of which is dense in X , such that f , restricted to each U i , is constant. A number of characterizations of DC-spaces are given, which lead to an algebraic generalization of the concept, which, in turn, permits analysis of DC-spaces in the language of archimedean f -algebras. One is led naturally to the notion of an almost DC-space (in which the densely constant functions...

Almost maximal topologies on groups

Yevhen Zelenyuk (2016)

Fundamenta Mathematicae

Let G be a countably infinite group. We show that for every finite absolute coretract S, there is a regular left invariant topology on G whose ultrafilter semigroup is isomorphic to S. As consequences we prove that (1) there is a right maximal idempotent in βG∖G which is not strongly right maximal, and (2) for each combination of the properties of being extremally disconnected, irresolvable, and nodec, except for the combination (-,-,+), there is a corresponding regular almost maximal left invariant...

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