Coarser connected topologies

AbstractWe investigate which spaces have coarser connected topologies. If in a collectionwise normal space X, the density equals the extent, which is attained and at least c, then X has a coarser connected collectionwise normal topology. In the previous sentence, the separation property collectionwise normal can be replaced by other separation properties—for example, Hausdorff, Urysohn, regular, metrizable. A zero-dimensional metrizable space X of density at least c has a coarser connected metrizable topology. A non-H-closed Hausdorff space with a σ-locally finite base has a coarser connected Hausdorff topology. We give necessary conditions and sufficient conditions for an ordinal to have a coarser connected Urysohn topology. In particular, every indecomposible ordinal of countable cofinality has a coarser connected topology. We present a nowhere locally compact Hausdorff space X with no coarser connected Hausdorff topology, yet X is dense in a connected Hausdorff space Y

Topics in uniform continuity

This paper collects results and open problems concerning several classes of functions that generalize uniform continuity in various ways, including those metric spaces (generalizing Atsuji spaces) where all continuous functions have the property of being close to uniformly continuous

Coarser connected topologies and non-normality points

We investigate two topics, coarser connected topologies and non-normality points. The motivating question in the first topic is: When does a space have a coarser connected topology with a nice topological property? We will discuss some results when the property is Hausdorff and prove that if X is a non-compact metric space that has weight at least the cardinality of the continuum, then it has a coarser connected metrizable topology. The second topic is concerned with the following question: When is a point of the Stone-Cech remainder of a space a non-normality point of the remainder? We will discuss the question in the case that X is a discrete space and then when X is a metric space without isolated points. We show that under certain set-theoretic conditions, if X is a locally compact metric space without isolated points then every point in the Stone-Cech remainder is a non-normality point of the remainder

Hindman's finite sums theorem and its application to topologizations of algebras

The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent ultrafilters in ultrafilter extensions of semigroups. The second, main part of the paper is devoted to the topologizability problem of a wide class of algebraic structures called polyrings; this class includes Abelian groups, rings, modules, algebras over a ring, differential rings, and others. We show that the Zariski topology on such an algebra is always non-discrete. Actually, a much stronger fact holds: if $K$ is an infinite polyring, $n$ a natural number, and a map $F$ of $K^n$ into $K$ is defined by a term in $n$ variables, then $F$ is a closed nowhere dense subset of the space $K^{n+1}$ with its Zariski topology. In particular, $K^n$ is a closed nowhere dense subset of $K^{n+1}$. The proof essentially uses a multidimensional version of Hindman's finite sums theorem established by Bergelson and Hindman. The third part of the paper lists several problems concerning topologization of various algebraic structures, their Zariski topologies, and some related questions. This paper is an extended version of the lecture at Journ\'ees sur les Arithm\'etiques Faibles 36: \`a l'occasion du 70\`eme anniversaire de Yuri Matiyasevich, delivered on 7th July, 2017, in Saint Petersburg.Comment: The main result of the paper, Theorem 2.4.1, was proved around 2010 but not published until 2017 though presented at several seminars and conferences, e.g. Colloquium Logicum 2012 in Paderborn, and included in author's course lectured at the Steklov Mathematical Institute in 201

Coarser connected metrizable topologies

AbstractWe show that every metric space, X, with w(X)⩾c has a coarser connected metrizable topology

Polyfolds: A First and Second Look

Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding commonalities in the analytic framework for a variety of geometric elliptic PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to systematically address the common difficulties of compactification and transversality with a new notion of smoothness on Banach spaces, new local models for differential geometry, and a nonlinear Fredholm theory in the new context. We shine meta-mathematical light on the bigger picture and core ideas of this theory. In addition, we compiled and condensed the core definitions and theorems of polyfold theory into a streamlined exposition, and outline their application at the example of Morse theory.Comment: 62 pages, 2 figures. Example 2.1.3 has been modified. Final version, to appear in the EMS Surv. Math. Sc