Expand all Collapse all | Results 26 - 50 of 50 |
26. CMB 2010 (vol 54 pp. 180)
Additive Families of Low Borel Classes and Borel Measurable Selectors
An important conjecture in the theory of Borel sets in non-separable
metric spaces is whether any point-countable Borel-additive family in
a complete metric space has a $\sigma$-discrete refinement. We confirm the conjecture for
point-countable $\mathbf\Pi_3^0$-additive families, thus generalizing results of
R. W. Hansell and the first author. We apply this result to the
existence of Borel measurable selectors for multivalued mappings of
low Borel complexity, thus answering in the affirmative a particular
version of a question of J. Kaniewski and R. Pol.
Keywords:$\sigma$-discrete refinement, Borel-additive family, measurable selection Categories:54H05, 54E35 |
27. CMB 2010 (vol 53 pp. 719)
A Continuous Extension Operator for Convex Metrics
We consider the problem of simultaneous extension of continuous
convex metrics defined on subcontinua of a Peano continuum. We prove
that there is an extension operator for convex metrics that is
continuous with respect to the uniform topology.
Categories:54E35, 54C20, 54E40 |
28. CMB 2010 (vol 53 pp. 629)
Asymptotic Dimension of Proper CAT(0) Spaces that are Homeomorphic to the Plane In this paper, we investigate
a proper CAT(0) space $(X,d)$
that is homeomorphic to $\mathbb R^2$ and
we show that the asymptotic dimension $\operatorname{asdim} (X,d)$ is
equal to $2$.
Keywords:asymptotic dimension, CAT(0) space, plane Categories:20F69, 54F45, 20F65 |
29. CMB 2010 (vol 53 pp. 438)
Near-Homeomorphisms of Nöbeling Manifolds We characterize maps between $n$-dimensional NÃ¶beling manifolds that can be approximated by homeomorphisms.
Keywords:n-dimensional Nöbeling manifold, Z-set unknotting, near-homeomorphism Categories:55M10, 54F45 |
30. CMB 2010 (vol 53 pp. 360)
Separating H-sets by Open Sets In an H-closed, Urysohn space, disjoint H-sets can be separated by disjoint open sets. This is not true for an arbitrary H-closed space even if one of the H-sets is a point. In this paper, we provide a systematic study of those spaces in which disjoint H-sets can be separated by disjoint open sets.
Keywords:H-set, H-closed, θ-continuous Categories:54C08, 54D10, 54D15 |
31. CMB 2010 (vol 53 pp. 286)
Orders of π-Bases We extend the scope of B. Shapirovskii's results on the order of $\pi$-bases in compact spaces and answer some questions of V. Tkachuk.
Keywords:Shapirovskii π-base, point-countable π-base, free sequences, canonical form for ordinals Categories:54A25, 03E10, 03E75, 54A35 |
32. CMB 2009 (vol 52 pp. 544)
Intuitionistic Fuzzy $\gamma$-Continuity This paper introduces the concepts of
fuzzy $\gamma$-open sets and fuzzy $\gamma$-continuity
in intuitionistic fuzzy topological spaces. After defining the fundamental
concepts of intuitionistic fuzzy sets and intuitionistic fuzzy topological
spaces, we present intuitionistic fuzzy $\gamma$-open sets and
intuitionistic fuzzy $\gamma$-continuity and other results related
topological concepts.
Keywords:intuitionistic fuzzy set, intuitionistic fuzzy point, intuitionistic fuzzy topological space, intuitionistic fuzzy $\gamma$-open set, intuitionistic fuzzy $\gamma$-\continuity, intuitionistic fuzzy $\gamma$-closure ($\gamma$-interior) Categories:54A40, 54A20, 54F99 |
33. CMB 2009 (vol 52 pp. 295)
On Functions Whose Graph is a Hamel Basis, II We say that a function $h \from \real \to \real$ is a Hamel function
($h \in \ham$) if $h$, considered as a subset of $\real^2$, is a Hamel
basis for $\real^2$. We show that $\A(\ham)\geq\omega$, \emph{i.e.,} for
every finite $F \subseteq \real^\real$ there exists $f\in\real^\real$
such that $f+F \subseteq \ham$. From the previous work of the author
it then follows that $\A(\ham)=\omega$.
Keywords:Hamel basis, additive, Hamel functions Categories:26A21, 54C40, 15A03, 54C30 |
34. CMB 2008 (vol 51 pp. 570)
Amsterdam Properties of $C_p(X)$ Imply Discreteness of $X$ We prove, among other things, that if $C_p(X)$ is
subcompact in the sense of de Groot, then the space $X$ is
discrete. This generalizes a series of previous results on
completeness properties of function spaces.
Keywords:regular filterbase, subcompact space, function space, discrete space Categories:54B10, 54C05, 54D30 |
35. CMB 2008 (vol 51 pp. 413)
Big Ramsey Degrees and Divisibility in Classes of Ultrametric Spaces Given a countable set $S$ of positive reals, we study
finite-dimensional Ramsey-theoretic properties of the countable
ultrametric Urysohn space $\textbf{Q} _S$ with distances in $S$.
Keywords:Ramsey theory, Urysohn metric spaces, ultrametric spaces Categories:05C50, 54E35 |
36. CMB 2008 (vol 51 pp. 310)
Relative Homotopy in Relational Structures The homotopy groups of a finite partially ordered set (poset) can be
described entirely in the context of posets, as shown in a paper by
B. Larose and C. Tardif.
In this paper we describe the relative version of such a
homotopy theory, for pairs $(X,A)$ where $X$ is a poset and $A$ is a
subposet of $X$. We also prove some theorems on the relevant version
of the notion of weak homotopy equivalences for maps of pairs of such
objects. We work in the category of reflexive binary relational
structures which contains the posets as in the work of Larose and
Tardif.
Keywords:binary reflexive relational structure, relative homotopy group, exact sequence, locally finite space, weak homotopy equivalence Categories:55Q05, 54A05;, 18B30 |
37. CMB 2005 (vol 48 pp. 614)
On Finite-to-One Maps Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective
map of metrizable spaces such that $\dim Y\leq m$. It is shown that
for every positive integer $p$ with $ p\leq m+k+1$ there exists a
dense $G_{\delta}$-subset ${\mathcal H}(k,m,p)$ of $C(X,\uin^{k+p})$
with the source limitation topology such that each fiber of
$f\triangle g$, $g\in{\mathcal H}(k,m,p)$, contains at most
$\max\{k+m-p+2,1\}$ points. This result
provides a proof the following conjectures of
S. Bogatyi, V. Fedorchuk and J. van Mill.
Let $f\colon X\to Y$ be a $k$-dimensional map between compact
metric spaces with $\dim Y\leq m$. Then:
\begin{inparaenum}[\rm(1)]
\item there exists a map
$h\colon X\to\uin^{m+2k}$ such that $f\triangle h\colon X\to
Y\times\uin^{m+2k}$ is 2-to-one provided $k\geq 1$;
\item there exists a
map $h\colon X\to\uin^{m+k+1}$ such that $f\triangle h\colon X\to
Y\times\uin^{m+k+1}$ is $(k+1)$-to-one.
\end{inparaenum}
Keywords:finite-to-one maps, dimension, set-valued maps Categories:54F45, 55M10, 54C65 |
38. CMB 2005 (vol 48 pp. 195)
On Suslinian Continua A continuum is said to be Suslinian if it does not contain uncountably
many mutually exclusive nondegenerate subcontinua. We prove that
Suslinian continua are perfectly normal and rim-metrizable. Locally
connected Suslinian continua have weight at most $\omega_1$ and under
appropriate set-theoretic conditions are metrizable. Non-separable
locally connected Suslinian continua are rim-finite on some open set.
Keywords:Suslinian continuum, Souslin line, locally connected, rim-metrizable,, perfectly normal, rim-finite Categories:54F15, 54D15, 54F50 |
39. CMB 2003 (vol 46 pp. 291)
A Coincidence Theorem for Holomorphic Maps to $G/P$ The purpose of this note is to extend to an arbitrary generalized Hopf
and Calabi-Eckmann manifold the following result of Kalyan Mukherjea:
Let $V_n = \mathbb{S}^{2n+1} \times \mathbb{S}^{2n+1}$ denote a
Calabi-Eckmann manifold. If $f,g \colon V_n \longrightarrow
\mathbb{P}^n$ are any two holomorphic maps, at least one of them being
non-constant, then there exists a coincidence: $f(x)=g(x)$ for some
$x\in V_n$. Our proof involves a coincidence theorem for holomorphic
maps to complex projective varieties of the form $G/P$ where $G$ is
complex simple algebraic group and $P\subset G$ is a maximal parabolic
subgroup, where one of the maps is dominant.
Categories:32H02, 54M20 |
40. CMB 2001 (vol 44 pp. 266)
Extension of Maps to Nilpotent Spaces We show that every compactum has cohomological dimension $1$ with respect
to a finitely generated nilpotent group $G$ whenever it has cohomological
dimension $1$ with respect to the abelianization of $G$. This is applied
to the extension theory to obtain a cohomological dimension theory condition
for a finite-dimensional compactum $X$ for extendability of every map from
a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely
generated homotopy groups over all of $X$.
Keywords:cohomological dimension, extension of maps, nilpotent group, nilpotent space Categories:55M10, 55S36, 54C20, 54F45 |
41. CMB 2001 (vol 44 pp. 80)
Constructing Compacta of Different Extensional Dimensions Applying the Sullivan conjecture we construct compacta of certain
cohomological and extensional dimensions.
Keywords:cohomological dimension, Eilenberg-MacLane complexes, Sullivan conjecture Categories:55M10, 54F45, 55U20 |
42. CMB 2000 (vol 43 pp. 208)
Extensions of Continuous and Lipschitz Functions We show a result slightly more general than the following. Let $K$
be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a
lower semi-continuous metric on $K$. Then each continuous function
$f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on
$K$ which is Lipschitz in $d$. The extension has the same supremum
norm and the same Lipschitz constant.
As a corollary we get that a Banach space $X$ is reflexive if and only
if each bounded, weakly continuous and norm Lipschitz function
defined on a weakly closed subset of $X$ admits a weakly continuous,
norm Lipschitz extension defined on the entire space $X$.
Keywords:extension, continous, Lipschitz, Banach space Categories:54C20, 46B10 |
43. CMB 1999 (vol 42 pp. 190)
Topological Quantum Field Theory and Strong Shift Equivalence Given a TQFT in dimension $d+1,$ and an infinite cyclic covering of
a closed $(d+1)$-dimensional manifold $M$, we define an invariant
taking values in a strong shift equivalence class of matrices. The
notion of strong shift equivalence originated in R.~Williams' work
in symbolic dynamics. The Turaev-Viro module associated to a TQFT
and an infinite cyclic covering is then given by the Jordan form of
this matrix away from zero. This invariant is also defined if the
boundary of $M$ has an $S^1$ factor and the infinite cyclic cover
of the boundary is standard. We define a variant of a TQFT
associated to a finite group $G$ which has been studied by Quinn.
In this way, we recover a link invariant due to D.~Silver and
S.~Williams. We also obtain a variation on the Silver-Williams
invariant, by using the TQFT associated to $G$ in its unmodified form.
Keywords:knot, link, TQFT, symbolic dynamics, shift equivalence Categories:57R99, 57M99, 54H20 |
44. CMB 1999 (vol 42 pp. 13)
Dow's Principle and $Q$-Sets A $Q$-set is a set of reals every subset of which is a relative
$G_\delta$. We investigate the combinatorics of $Q$-sets and
discuss a question of Miller and Zhou on the size $\qq$ of the smallest
set of reals which is not a $Q$-set. We show in particular that various
natural lower bounds for $\qq$ are consistently strictly smaller than
$\qq$.
Keywords:$Q$-set, cardinal invariants of the continuum, pseudointersection number, $\MA$($\sigma$-centered), Dow's principle, almost disjoint family, almost disjointness principle, iterated forcing Categories:03E05, 03E35, 54A35 |
45. CMB 1998 (vol 41 pp. 348)
Characterizing continua by disconnection properties We study Hausdorff continua in which every set of certain
cardinality contains a subset which disconnects the space. We show
that such continua are rim-finite. We give characterizations of
this class among metric continua. As an application of our
methods, we show that continua in which each countably infinite set
disconnects are generalized graphs. This extends a result of
Nadler for metric continua.
Keywords:disconnection properties, rim-finite continua, graphs Categories:54D05, 54F20, 54F50 |
46. CMB 1998 (vol 41 pp. 245)
The normality in products with a countably compact factor It is known that the product $\omega_1 \times X$ of
$\omega_1$ with an $M_1$-space may be nonnormal. In this paper we
prove that the product $\kappa \times X$ of an uncountable regular
cardinal $\kappa$ with a paracompact semi-stratifiable space is normal
if{f} it is countably paracompact. We also give a sufficient
condition under which the product of a normal space with a paracompact
space is normal, from which many theorems involving such a product
with a countably compact factor can be derived.
Categories:54B19, 54D15, 54D20 |
47. CMB 1997 (vol 40 pp. 395)
$D$-spaces and resolution A space $X$ is a $D$-space if, for every neighborhood assignment $f$
there is a closed discrete set $D$ such that $\bigcup{f(D)}=X$. In this
paper we give some necessary conditions and some sufficient conditions
for a resolution of a topological space to be a $D$-space. In particular,
if a space $X$ is resolved at each $x\in X$ into a $D$-space $Y_x$ by
continuous mappings $f_x\colon X-\{{x}\} \rightarrow Y_x$, then the
resolution is a $D$-space if and only if $\bigcup{\{{x}\}}\times \Bd(Y_x)$
is a $D$-space.
Keywords:$D$-space, neighborhood assignment, resolution, boundary Categories:54D20, 54B99, 54D10, 54D30 |
48. CMB 1997 (vol 40 pp. 448)
Stable index pairs for discrete dynamical systems A new shorter proof of the existence of index pairs for discrete
dynamical systems is given. Moreover, the index pairs defined in
that proof are stable with respect to small perturbations of the
generating map. The existence of stable index pairs was previously
known in the case of diffeomorphisms and flows generated by smooth
vector fields but it was an open question in the general discrete
case.
Categories:54H20, 54C60, 34C35 |
49. CMB 1997 (vol 40 pp. 422)
On compact separable radial spaces If ${\cal A} $ and ${\cal B}$ are disjoint ideals on $\omega$, there is
a {\it tower preserving\/} $\sigma$-centered forcing which introduces a
subset of $\omega$ which meets every infinite member of ${\cal A}$ in
an infinite set and is almost disjoint from every member of ${\cal B}$.
We can then produce a model in which all compact separable radial
spaces are Fr\'echet, thus answering a question of P.~Nyikos. The
question of the existence of compact ccc radial spaces which are not
Fr\'echet was first asked by Chertanov (see \cite{Ar78}).
Category:54D30 |
50. CMB 1997 (vol 40 pp. 39)
On projective $Z$-frames This paper deals with the projective objects in the category of all
$Z$-frames, where the latter is a common generalization of
different types of frames. The main result obtained here is that a
$Z$-frame is ${\bf E}$-projective if and only if it is stably
$Z$-continuous, for a naturally arising collection ${\bf E}$ of morphisms.
Categories:06D05, 54D10, 18D15 |