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26. CMB 2010 (vol 53 pp. 719)

Stasyuk, I.; Tymchatyn, E. D.
 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

27. CMB 2010 (vol 53 pp. 629)

Chinen, Naotsugu; Hosaka, Tetsuya
 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, planeCategories:20F69, 54F45, 20F65

28. CMB 2010 (vol 53 pp. 438)

Chigogidze, A.; Nagórko, A.
 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-homeomorphismCategories:55M10, 54F45

29. CMB 2010 (vol 53 pp. 360)

Porter, Jack; Tikoo, Mohan
 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-continuousCategories:54C08, 54D10, 54D15

30. CMB 2010 (vol 53 pp. 286)

Gorelic, Isaac
 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 ordinalsCategories:54A25, 03E10, 03E75, 54A35

31. 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

32. 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 functionsCategories:26A21, 54C40, 15A03, 54C30

33. 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 spaceCategories:54B10, 54C05, 54D30

34. CMB 2008 (vol 51 pp. 413)

Thé, L. Nguyen Van
 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 spacesCategories:05C50, 54E35

35. 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 equivalenceCategories:55Q05, 54A05;, 18B30

36. CMB 2005 (vol 48 pp. 614)

Tuncali, H. Murat; Valov, Vesko
 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 mapsCategories:54F45, 55M10, 54C65

37. 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 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-finiteCategories:54F15, 54D15, 54F50

38. 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 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

39. 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 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 spaceCategories:55M10, 55S36, 54C20, 54F45

40. CMB 2001 (vol 44 pp. 80)

Levin, Michael
 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 conjectureCategories:55M10, 54F45, 55U20

41. CMB 2000 (vol 43 pp. 208)

Matoušková, Eva
 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 spaceCategories:54C20, 46B10

42. CMB 1999 (vol 42 pp. 190)

Gilmer, Patrick M.
 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 equivalenceCategories:57R99, 57M99, 54H20

43. CMB 1999 (vol 42 pp. 13)

Brendle, Jörg
 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 forcingCategories:03E05, 03E35, 54A35

44. CMB 1998 (vol 41 pp. 348)

Tymchatyn, E. D.; Yang, Chang-Cheng
 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, graphsCategories:54D05, 54F20, 54F50

45. CMB 1998 (vol 41 pp. 245)

Yang, Lecheng
 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

46. CMB 1997 (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, boundaryCategories:54D20, 54B99, 54D10, 54D30

47. CMB 1997 (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

48. CMB 1997 (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

49. CMB 1997 (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
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