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Asymmetric tie-points and almost clopen subsets of *

Alan S. Dow, Saharon Shelah (2018)

Commentationes Mathematicae Universitatis Carolinae

A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of * . One especially important application, due to Veličković, was to the existence of nontrivial involutions on * . A tie-point of * has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost clopen set...

Atomic compactness for reflexive graphs

Christian Delhommé (1999)

Fundamenta Mathematicae

A first order structure with universe M is atomic compact if every system of atomic formulas with parameters in M is satisfiable in provided each of its finite subsystems is. We consider atomic compactness for the class of reflexive (symmetric) graphs. In particular, we investigate the extent to which “sparse” graphs (i.e. graphs with “few” vertices of “high” degree) are compact with respect to systems of atomic formulas with “few” unknowns, on the one hand, and are pure restrictions of their...

Aull-paracompactness and strong star-normality of subspaces in topological spaces

Kaori Yamazaki (2004)

Commentationes Mathematicae Universitatis Carolinae

We prove for a subspace Y of a T 1 -space X , Y is (strictly) Aull-paracompact in X and Y is Hausdorff in X if and only if Y is strongly star-normal in X . This result provides affirmative answers to questions of A.V. Arhangel’skii–I.Ju. Gordienko [3] and of A.V. Arhangel’skii [2].

Axiom T D and the Simmons sublocale theorem

Jorge Picado, Aleš Pultr (2019)

Commentationes Mathematicae Universitatis Carolinae

More precisely, we are analyzing some of H. Simmons, S. B. Niefield and K. I. Rosenthal results concerning sublocales induced by subspaces. H. Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom T D for the relation of certain degrees of scatteredness but did not emphasize its role in the relation between sublocales and subspaces. S. B. Niefield and K. I. Rosenthal just mention this axiom in a remark about Simmons’ result. In this...

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