26. CMB 2011 (vol 54 pp. 607)
 Camargo, Javier

Lightness of Induced Maps and Homeomorphisms
An example is given of a map $f$ defined between arcwise connected continua such that $C(f)$ is light and
$2^{f}$ is not light, giving a negative answer to a question of Charatonik and Charatonik. Furthermore, given a positive
integer $n$, we study when the lightness of the induced map $2^{f}$ or $C_n(f)$ implies that $f$ is a
homeomorphism. Finally, we show a result in relation with the lightness of $C(C(f))$.
Keywords:light maps, induced maps, continua, hyperspaces Categories:54B20, 54E40 

27. CMB 2011 (vol 54 pp. 244)
 Daniel, D. ; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.

Homogeneous Suslinian Continua
A continuum is said to be Suslinian if it does not
contain uncountably many
mutually exclusive nondegenerate subcontinua. Fitzpatrick and
Lelek have shown that a metric Suslinian continuum $X$ has the
property that the set of points at which $X$ is connected im
kleinen is dense in $X$. We extend their result to Hausdorff Suslinian continua
and obtain a number of corollaries. In particular, we prove that a homogeneous,
nondegenerate, Suslinian continuum is a simple closed curve and that each separable,
nondegenerate, homogenous, Suslinian continuum is metrizable.
Keywords:connected im kleinen, homogeneity, Suslinian, locally connected continuum Categories:54F15, 54C05, 54F05, 54F50 

28. CMB 2011 (vol 54 pp. 302)
29. CMB 2010 (vol 54 pp. 193)
 Bennett, Harold; Lutzer, David

Measurements and $G_\delta$Subsets of Domains
In this paper we study domains, Scott
domains, and the existence of measurements. We
use a space created by D.~K. Burke to show that
there is a Scott domain $P$ for which $\max(P)$ is
a $G_\delta$subset of $P$ and yet no measurement
$\mu$ on $P$ has $\ker(\mu) = \max(P)$. We also
correct a mistake in the literature asserting that
$[0, \omega_1)$ is a space of this type. We show
that if $P$ is a Scott domain and $X \subseteq
\max(P)$ is a $G_\delta$subset of $P$, then $X$
has a $G_\delta$diagonal and is weakly
developable. We show that if $X \subseteq
\max(P)$ is a $G_\delta$subset of $P$, where
$P$ is a domain but perhaps not a Scott domain,
then $X$ is domainrepresentable,
firstcountable, and is the union of dense,
completely metrizable subspaces. We also
show that there is a domain $P$ such that
$\max(P)$ is the usual space of countable
ordinals and is a $G_\delta$subset of $P$ in
the Scott topology. Finally we show that the
kernel of a measurement on a Scott domain can
consistently be a normal, separable,
nonmetrizable Moore space.
Keywords:domainrepresentable, Scottdomainrepresentable, measurement, Burke's space, developable spaces, weakly developable spaces, $G_\delta$diagonal, Äechcomplete space, Moore space, $\omega_1$, weakly developable space, sharp base, AFcomplete Categories:54D35, 54E30, 54E52, 54E99, 06B35, 06F99 

30. CMB 2010 (vol 54 pp. 270)
 Dow, Alan

Sequential Order Under PFA
It is shown that it follows from PFA
that there is no
compact scattered space of height greater than $\omega$
in which the sequential order and the scattering heights coincide.
Keywords:sequential order, scattered spaces, PFA Categories:54D55, 03E05, 03E35, 54A20 

31. CMB 2010 (vol 54 pp. 180)
 Spurný, J.; Zelený, M.

Additive Families of Low Borel Classes and Borel Measurable Selectors
An important conjecture in the theory of Borel sets in nonseparable
metric spaces is whether any pointcountable Boreladditive family in
a complete metric space has a $\sigma$discrete refinement. We confirm the conjecture for
pointcountable $\mathbf\Pi_3^0$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, Boreladditive family, measurable selection Categories:54H05, 54E35 

32. CMB 2010 (vol 53 pp. 719)
33. CMB 2010 (vol 53 pp. 629)
34. CMB 2010 (vol 53 pp. 438)
35. CMB 2010 (vol 53 pp. 286)
 Gorelic, Isaac

Orders of πBases
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 πbase, pointcountable πbase, free sequences, canonical form for ordinals Categories:54A25, 03E10, 03E75, 54A35 

36. CMB 2010 (vol 53 pp. 360)
 Porter, Jack; Tikoo, Mohan

Separating Hsets by Open Sets
In an Hclosed, Urysohn space, disjoint Hsets can be separated by disjoint open sets. This is not true for an arbitrary Hclosed space even if one of the Hsets is a point. In this paper, we provide a systematic study of those spaces in which disjoint Hsets can be separated by disjoint open sets.
Keywords:Hset, Hclosed, θcontinuous Categories:54C08, 54D10, 54D15 

37. CMB 2009 (vol 52 pp. 544)
 Hanafy, I. M.

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 

38. CMB 2009 (vol 40 pp. 395)
 Boudhraa, Zineddine

$D$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 

39. CMB 2009 (vol 52 pp. 295)
 P{\l}otka, Krzysztof

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 

40. CMB 2009 (vol 40 pp. 448)
 Kaczynski, Tomasz; Mrozek, Marian

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 

41. CMB 2009 (vol 40 pp. 422)
 Dow, Alan

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 

42. CMB 2009 (vol 40 pp. 39)
 Zhao, Dongsheng

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 

43. CMB 2008 (vol 51 pp. 570)
 Lutzer, D. J.; Mill, J. van; Tkachuk, V. V.

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 

44. CMB 2008 (vol 51 pp. 413)
45. CMB 2008 (vol 51 pp. 310)
 Witbooi, P. J.

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 

46. CMB 2005 (vol 48 pp. 614)
 Tuncali, H. Murat; Valov, Vesko

On FinitetoOne 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
$\max\{k+mp+2,1\}$ 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 2toone 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 $(k+1)$toone.
\end{inparaenum}
Keywords:finitetoone maps, dimension, setvalued maps Categories:54F45, 55M10, 54C65 

47. CMB 2005 (vol 48 pp. 195)
 Daniel, D.; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.

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 rimmetrizable. Locally
connected Suslinian continua have weight at most $\omega_1$ and under
appropriate settheoretic conditions are metrizable. Nonseparable
locally connected Suslinian continua are rimfinite on some open set.
Keywords:Suslinian continuum, Souslin line, locally connected, rimmetrizable,, perfectly normal, rimfinite Categories:54F15, 54D15, 54F50 

48. CMB 2003 (vol 46 pp. 291)
 Sankaran, Parameswaran

A Coincidence Theorem for Holomorphic Maps to $G/P$
The purpose of this note is to extend to an arbitrary generalized Hopf
and CalabiEckmann manifold the following result of Kalyan Mukherjea:
Let $V_n = \mathbb{S}^{2n+1} \times \mathbb{S}^{2n+1}$ denote a
CalabiEckmann manifold. If $f,g \colon V_n \longrightarrow
\mathbb{P}^n$ are any two holomorphic maps, at least one of them being
nonconstant, 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 

49. CMB 2001 (vol 44 pp. 266)
 Cencelj, M.; Dranishnikov, A. N.

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 finitedimensional 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 

50. CMB 2001 (vol 44 pp. 80)