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51. 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, boundary
Categories:54D20, 54B99, 54D10, 54D30

52. 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

53. 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}).


54. 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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