Some Applications of the Poincaré–Bendixson Theorem

We consider a $${\mathcal {C}}^1$$ C 1 vector field X defined on an open subset U of the plane with compact closure. If X has no singular points and if U is simply connected, a weak version of the Poincaré–Bendixson theorem says that the limit sets of X in U are empty but that one can define non empty extended limit sets contained in the boundary of U. We give an elementary proof of this result, independent of the classical Poincaré–Bendixson theorem. A trapping triangle $${\mathcal {T}}$$ T based at p, for a $${\mathcal {C}}^1$$ C 1 vector field X defined on an open subset $${\mathcal {U}}$$ U of the plane, is a topological triangle with a corner at a point p located on the boundary $$\partial {\mathcal {U}}$$ ∂ U and a good control of the tranversality of X along the sides. The principal application of the weak Poincaré–Bendixson theorem is that a trapping triangle at p contains a separatrix converging toward the point p. This does not depend on the properties of X along $$\partial {\mathcal {U}}$$ ∂ U . For instance, X could be non differentiable at p, as in the example presented in the last section.

Local behavior of mappings of metric spaces with branching

The local behavior of mappings with the inverse Poletsky inequality between metric spaces is studied. The case where one of the spaces satisfies the condition of weak sphericalization, is similar to the Riemannian sphere (extended Euclidean space), and is locally linearly connected under a mapping is considered. It is proved that the equicontinuity of the corresponding families of mappings of two domains, one of which is a domain with a weakly flat boundary, and another one is a fixed domain with a compact closure, the corresponding weight in the main inequality being supposed to be integrable.

On cardinalities and compact closures

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>We show that there exists a Hausdorff topology on the set </span><span>R </span><span>of real numbers such that a subset </span><span>A </span><span>of </span><span>R </span><span>has compact closure if and only if </span><span>A </span><span>is countable. More generally, given any set </span><span>X </span><span>and any infinite set </span><span>S</span><span>, we prove that there exists a Hausdorff topology on </span><span>X </span><span>such that a subset </span><span>A </span><span>of </span><span>X </span><span>has compact closure if and only if the cardinality of </span><span>A </span><span>is less than or equal to that of </span><span>S</span><span>. When we attempt to replace “than than or equal to” in the preceding statement with “strictly less than,” the situation is more delicate; we show that the theorem extends to this case when </span><span>S </span><span>has regular cardinality but can fail when it does not. This counterexample shows that not every bornology is a bornology of compact closure. These results lie in the intersection of analysis, general topology, and set theory. </span></p></div></div></div>

Two classes of locally compact sober spaces

We deal with two classes of locally compact sober spaces, namely, the class of locally spectral coherent spaces and the class of spaces in which every point has a closed spectral neighborhood (CSN-spaces, for short). We prove that locally spectral coherent spaces are precisely the coherent sober spaces with a basis of compact open sets. We also prove that CSN-spaces are exactly the locally spectral coherent spaces in which every compact open set has a compact closure.

Ultrafilters on the natural numbers

We study the problem of existence and generic existence of ultrafilters on ω. We prove a conjecture of Jörg Brendle's showing that there is an ultrafilter that is countably closed but is not an ordinal ultrafilter under CH. We also show that Canjar's previous partial characterization of the generic existence of Q-points is the best that can be done. More simply put, there is no normal cardinal invariant equality that fully characterizes the generic existence of Q-points. We then sharpen results on generic existence with the introduction of σ-compact ultrafilters. We show that the generic existence of said ultrafilters is equivalent to . This result, taken along with our result that there exists a Kσ, non-countably closed ultrafilter under CH, expands the size of the class of ultrafilters that were known to fit this description before. From the core of the proof, we get a new result on the cardinal invariants of the continuum, i.e., the cofinality of the sets with σ-compact closure is .

$FC^-$-elements in totally disconnected groups and automorphisms of infinite graphs

An element in a topological group is called an $\mathrm{FC}^-$-element if its conjugacy class has compact closure. The $\mathrm{FC}^-$-elements form a normal subgroup. In this note it is shown that in a compactly generated totally disconnected locally compact group this normal subgroup is closed. This result answers a question of Ghahramani, Runde and Willis. The proof uses a result of Trofimov about automorphism groups of graphs and a graph theoretical interpretation of the condition that the group is compactly generated.

Uniform Distribution in Model Sets

We give a new measure-theoretical proof of the uniform distribution property of points in model sets (cut and project sets). Each model set comes as a member of a family of related model sets, obtained by joint translation in its ambient (the ‘physical’) space and its internal space. We prove, assuming only that the window defining themodel set ismeasurable with compact closure, that almost surely the distribution of points in any model set from such a family is uniform in the sense of Weyl, and almost surely the model set is pure point diffractive.