论文标题
紧凑空间上的缩回和封闭子集的家族
Families of retractions and families of closed subsets on compact spaces
论文作者
论文摘要
众所周知，Valdivia紧凑型空间的特征是特殊的缩回系列，称为$ r $ -skeleton（请参阅\ cite {kubis1}）。我们也知道，有$ r $ skeletons的紧凑型空间不是瓦尔迪维亚。在本文中，我们将研究$ r $ $ - 平面和特别的紧凑型空间子集的特殊家庭。我们证明，如果$ x $是零维紧凑空间，而$ \ {r_s：s \ inγ\} $是$ x $上的$ r $ -skeleton，则$ | r_s（x）| \leqΩ$对于γ$中的所有$ s \，然后$ x $具有由孤立点组成的密集子集。另外，我们为$ r $ skeleton提供条件，以便可以将这种$ r $ skeleton扩展到基本空间的Alexandroff副本上的$ r $ skeleton。 Valdivia紧凑型空间的标准定义是通过单位间隔的功率的$σ$产生的。在这个事实之后，我们通过将$ x $与$ x $与单位间隔的合适功率嵌入到$ x $上的$π$ - 骨骼以及一对$（\ mathcal {f}，φ）$中，其中$ \ nathcal {f} $是$ x $ $ x $ $ $ $ $ $ $ $ $ $ $ x $ $ x $ - mathcal {f} $ nimate $ x $ - $ $ $ - $ - $ - $ $ - $ - $ - $ - $ - $ - $ qumy-- $ - $ qum- niws and-ymone。这个新的概念概括了$σ$ - 产品的想法。 我们证明，紧凑的空间接受了伸缩式骨骼，如果它承认了$π$ - 骨骼。这种等效性允许给出一个新的证据，证明具有缩减骨骼的紧凑空间的乘积允许伸缩式骨骼（请参阅\ cite {cuth1}）。在\ cite {casa1}中，Corson紧凑型空间的特征是特殊的封闭子集。遵循此方向，我们介绍了弱$ c $ - 骨骼的概念，在某些条件下，瓦尔迪维亚紧凑型空间和带有$ r $ skeletons的紧凑空间的特征。
It is know that the Valdivia compact spaces can be characterized by a special family of retractions called $r$-skeleton (see \cite{kubis1}). Also we know that there are compact spaces with $r$-skeletons which are not Valdivia. In this paper, we shall study $r$-squeletons and special families of closed subsets of compact spaces. We prove that if $X$ is a zero-dimensional compact space and $\{r_s:s\in Γ\}$ is an $r$-skeleton on $X$ such that $|r_s(X)| \leq ω$ for all $s\in Γ$, then $X$ has a dense subset consisting of isolated points. Also we give conditions to an $r$-skeleton in order that this $r$-skeleton can be extended to an $r$-skeleton on the Alexandroff Duplicate of the base space. The standard definition of a Valdivia compact spaces is via a $Σ$-product of a power of the unit interval. Following this fact we introduce the notion of $π$-skeleton on a compact space $X$ by embedding $X$ in a suitable power of the unit interval together with a pair $(\mathcal{F},φ)$, where $\mathcal{F}$ is family of metric separable subspaces of $X$ and $φ$ an $ω$-monotone function which satisfy certain properties. This new notion generalize the idea of a $Σ$-product. We prove that a compact space admits a retractional-skeleton iff it admits a $π$-skeleton. This equivalence allows to give a new proof of the fact that the product of compact spaces with retractional-skeletons admits an retractional-skeleton (see \cite{cuth1}). In \cite{casa1}, the Corson compact spaces are characterized by a special family of closed subsets. Following this direction, we introduce the notion of weak $c$-skeleton which under certain conditions characterizes the Valdivia compact spaces and compact spaces with $r$-skeletons.
