ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Subset
Power set
Hyperpower set sequence
Hyperpower set
Hypersubset
Subset algebra
Set partition
Proper set partition
Definition D447
Open set partition
Formulation 0
Let $T = (X, \mathcal{T})$ be a D1106: Topological space.
A D11: Set $\mathcal{S} \subseteq \mathcal{P}(X)$ is an open partition of $X$ with respect to $T$ if and only if
(1) $\forall \, E, F \in \mathcal{S} \, (E \neq F \quad \Rightarrow \quad E \cap F = \emptyset)$
(2) $X = \cup \mathcal{S}$
(3) $\forall \, E \in \mathcal{S} : E \neq \emptyset$
(4) $\mathcal{S} \subseteq \mathcal{T}$
Children
Closed set partition