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 $(\{0,1{\}}^{\Lambda },{\tau }_0^{\Lambda })$ is discussed. Here, $\Lambda$ is any set with the cardinal number $\succ \aleph ,\{0,1{\}}^{\Lambda }=\{\phi :\Lambda \to \{0,1\}\},{\tau }_0$ is the discrete topology for $\{0,1\}$ and the topology ${\tau }_0^{\Lambda }$ for $\{0,1{\}}^{\Lambda }$ is the topology with the base $\beta =\{<G_{{\lambda }_1},\dots ,G_{{\lambda }_n}>;G_{{\lambda }_1}\in {\tau }_0,\dots ,G_{{\lambda }_n}\in {\tau }_0,\{{\lambda }_1,\dots ,{\lambda }_n\}\subset \Lambda ,n\in \mathbf{N}\}$ 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 $\{\phi :\Lambda \to \{0,1\};\phi ({\lambda }_1)\in E_{{\lambda }_1},\dots ,\phi ({\lambda }_n)\in E_{{\lambda }_n},\phi (\lambda )\in \{0,1\},\lambda \in \Lambda -\{{\lambda }_1,\dots ,{\lambda }_n\}\}$.

5 pages