Complexes and L-structures
Keiô Nagami, Koichi TSUDA · Journal of the Mathematical Society of Japan · 1981
The purpose of this paper is to study the simplicial complex $K$ with Whitehead topology from the point of view of L-structures.It will be shown that the capacity of $K$ to admit L-structures decreases as the dimension of $K$ increases.As a consequence we know that there is a gap between the class of $M_{1}$ -spaces and the class of weak L-spaces.Throughout the paper $K$ is a simplicial complex with Whitehead topology and simplexes of $K$ are so-called open ones.$K^{n}$ denotes the n-section of $K$ .As for terminology refer to the first author [3], [4] and [5].PROOF.When dim $K\leqq 0,$ $K$ is discrete and metrizable.Consider the case when dim $K=1$ .Let $H$ be an arbitrary closed set of $K$ .Let $\{s_{\alpha} : \alpha\in A\}$ be the set of l-simplexes of $K$ .Let $U$ be an open set of $K$ with $K^{0}-H\subset U\subset\overline{U}\subset K-H$ .For each $\alpha\in A$ , let $\mathcal{U}_{\alpha}$ be an approaching anti-cover of $(H\cap\overline{s}_{\alpha})\cup\partial s_{\alpha}$ in $\overline{s}_{\alpha}$ .Set $\mathcal{U}=(\cup\{\mathcal{U}_{\alpha} ; \alpha\in A\})\cup\{U\}$ .Then $\mathcal{U}$ is as can easily be seen an approaching anti-cover of $H$ in $K$ .That completes the proof.1.2.THEOREM.Let $K$ be the 2-section of an infinite full complex.Then $K$ is not an L-sPace.PROOF.Let $s$ be a l-simplex of $K$ and $\{s_{i} : i=1, 2, \}$ a sequence of distinct 2-simplexes of $K$ having $s$ as their common face.Let $P$ be an edimge point of $s$ and $\{p_{i}\}$ a sequence of points of $s$ with $\lim p_{i}=p$ .Let $\mathcal{U}$ be an arbitrary anti-cover of $\{p\}$ .Choose $U_{i}\in \mathcal{U}$ with $p_{i}\in U_{i}$ .Since $ U_{i}\cap s_{i} eq\emptyset$ for any $i$ , we can pick a point $q_{i}\in U_{i}\cap s_{i}$ for each $i$ .