A dendrite generated from {0,1}^{\Lambda}, Card\Lambda \succ \aleph

Abstract

The existence of a decomposition space with a dendritic structure of a topological space $(\{0,1\}^\Lambda ,\tau_{0}^\Lambda )$ is discussed. Here, $\Lambda $ is any set with the cardinal number $\succ \aleph , \{0,1\}^{\Lambda }=\{\varphi :\Lambda \rightarrow \{0,1\}\}, \tau_0$ is the discrete topology for $\{0,1\}$ and the topology $\tau_0^{\Lambda }$ for $\{0,1\}^\Lambda $ is the topology with the base $\beta =\{<G_{\lambda _1},\dots,G_{\lambda _n}>~;~G_{\lambda_1}\in \tau_0,\dots,G_{\lambda _n}\in \tau_0, \{\lambda _1,\dots,\lambda _n\}\subset \Lambda ,n\in {\bf N}\}$ where the notation $<E_{\lambda _1},\dots,E_{\lambda _n}>$ concerning the subset $E_{\lambda _i}, i=1,\dots,n$ of $\{0,1\}$ denotes the set $\{\varphi :\Lambda \rightarrow \{0,1\}~;~\varphi (\lambda _1)\in E_{\lambda _1},\dots,\varphi (\lambda _n)\in E_{\lambda _n}, \varphi (\lambda )\in \{0,1\}, \lambda \in \Lambda -\{\lambda _1,\dots,\lambda _n\}\}$.