A dendrite generated from {0,1}^Λ, CardΛ\succ \aleph

Abstract

The existence of a decomposition space with a dendritic structure of a topological space $(\{0,1\}^Λ,τ_{0}^Λ)$ is discussed. Here, $Λ$ is any set with the cardinal number $\succ \aleph , \{0,1\}^{Λ}=\{φ:Λ\rightarrow \{0,1\}\}, τ_0$ is the discrete topology for $\{0,1\}$ and the topology $τ_0^{Λ}$ for $\{0,1\}^Λ$ is the topology with the base $β=\{ ~;~G_{λ_1}\in τ_0,\dots,G_{λ_n}\in τ_0, \{λ_1,\dots,λ_n\}\subset Λ,n\in {\bf N}\}$ where the notation $ $ concerning the subset $E_{λ_i}, i=1,\dots,n$ of $\{0,1\}$ denotes the set $\{φ:Λ\rightarrow \{0,1\}~;~φ(λ_1)\in E_{λ_1},\dots,φ(λ_n)\in E_{λ_n}, φ(λ)\in \{0,1\}, λ\in Λ-\{λ_1,\dots,λ_n\}\}$.