On the Weight of Nowhere Dense Subsets in Compact Spaces

We study a new cardinal-valued invariant ndw(X) (calling it the nd-weight of X) of a topological space which is defined as the least upper bound of the weights of nowhere dense subsets of X. The main result is the proof of the inequality hl(X)ndw(X) for compact sets without isolated points ((hl is the hereditary Lindelof number). This inequality implies that a compact space without isolated points of countable nd-weight is completely normal. Assuming the continuum hypothesis, we construct an example of a nonmetrizable compact space of countable nd-weight without isolated points.     

Choice principles for special subsets of the real line

We study the role the axiom of choice plays in the existence of some special subsets of ? and its power set ?(?).     

Flows and Bisections in Cubic Graphs

A k‐weak bisection of a cubic graph G is a partition of the vertex‐set of G into two parts V₁ and V₂ of equal size, such that each connected component of the subgraph of G induced by () is a tree of at most vertices. This notion can be viewed as a relaxed version of nowhere‐zero flows, as it directly follows from old results of Jaeger that every cubic graph G with a circular nowhere‐zero r‐flow has a ‐weak bisection. In this article, we study problems related to the existence of k‐weak bisections. We believe that every cubic graph that has a perfect matching, other than the Petersen graph, admits a 4‐weak bisection and we present a family of cubic graphs with no perfect matching that do not admit such a bisection. The main result of this article is that every cubic graph admits a 5‐weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5‐flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs that do contain bridges.     

Effectively and Noneffectively Nowhere Simple Sets

R. Shore proved that every recursively enumerable (r. e.) set can be split into two (disjoint) nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice ? of all r. e. sets. Nowhere simple sets were further studied by D. Miller and J. Remmel, and we generalize some of their results. We characterize r. e. sets which can be split into two (non) effectively nowhere simple sets, and r. e. sets which can be split into two r. e. non-nowhere simple sets. We show that every r. e. set is either the disjoint union of two effectively nowhere simple sets or two noneffectively nowhere simple sets. We characterize r. e. sets whose every nontrivial splitting is into nowhere simple sets, and r. e. sets whose every nontrivial splitting is into effectively nowhere simple sets. R. Shore proved that for every effectively nowhere simple set A, the lattice L* (A) is effectively isomorphic to ?*, and that there is a nowhere simple set A such that L*(A) is not effectively isomorphic to ?*. We prove that every nonzero r. e. Turing degree contains a noneffectively nowhere simple set A with the lattice L*(A) effectively isomorphic to ?*. Mathematics Subject Classification: 03D25, 03D10.     

Hyperspaces of nowhere topologically complete spaces

It is proved that ifX is a connected locally continuumwise connected coanalytic nowhere topologically complete space, then the hyperspace 2 ^X of all nonempty compact subsets ofX is strongly universal in the class of all coanalytic spaces. Moreover, 2 ^X is homeomorphic to Π₂ ifX is a Baire space, and toQ∖Π₁ ifX contains a dense absoluteG _δ-setG ⊂X such that the intersectionG ∩U is connected for any open connectedU ⊂X. (Here Π₁, Π₁⊂X are the standard subsets of the Hilbert cubeQ absorbing for the classes of analytic and coanalytic spaces, respectively.) Similar results are obtained for higher projective classes. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 35–51, July, 1997. Translated by O. V. Sipacheva     

Nowhere Vanishing Torsion Closed Curves Always Hide Twice

Convexity may imply points of vanishing torsion as the spatial 4-vertex theorem shows. We state here that for a simple closed curve to have nowhere vanishing torsion, it must violate convexity hiding at least twice inside its convex hull. Both the 4-vertex and Lsquo;hiding-twice results are generalized by obtaining a relation between the number of vanishing torsion points (vertices) of a closed space curve and the number of its components inside its convex hull. We also comment on elastic curves.     

Snarks with given real flow numbers

We show that for each rational number r such that 4<r?5 there exist infinitely many cyclically 4‐edge‐connected cubic graphs of chromatic index 4 and girth at least 5—that is, snarks—whose flow number equals r. This answers a question posed by Pan and Zhu [Construction of graphs with given circular flow numbers, J Graph Theory 43 [2003], 304–318]. © 2011 Wiley Periodicals, Inc. J Graph Theory 68: 189‐201, 2011     

Remote points and the like in pointfree topology

A point p of β, where X is a Tychonoff space, is called a remote point if for any nowhere dense D ⊆, X, p ∉ cl_βx D. A subset S of X is called round if whenever the closure in βX of a zero-set of X contains S, then it is a neighborhood of S. The purpose of this paper is to study these notions in the pointfree context. In the process, we introduce N-homomorphisms and show how the Stone extension of an N-homomorphism transfers remote points back and forth.      

Topological groups in which each nowhere dense subset is closed

Assuming the validity of the combinatorial principlep=c, which follows from Martin's axiom, it is proved that an arbitrary nondiscrete metrizable group topology on an Abelian group can be strengthened to a nondiscrete group topology in which each nowhere dense subset is closed.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 207–211, August, 1998.