Complement of Irreducible Topological Subset is Prime Element
Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $X$ be an irreducible subset of $S$ such that:
- $\relcomp S X \in \tau$
Let $L = \struct {\tau, \preceq}$ be an inclusion ordered set of topology $\tau$.
Then $\relcomp S X$ is prime element in $L$.
Proof
Let $Y, Z \in \tau$ such that
- $Y \wedge Z \preceq \relcomp S X$
By definition of topological space:
- $Y \cap Z \in \tau$
By Meet in Inclusion Ordered Set:
- $Y \cap Z = Y \wedge Z$
By definition of inclusion ordered set:
- $Y \cap Z \subseteq \relcomp S X$
By Relative Complement inverts Subsets and Relative Complement of Relative Complement:
- $X \subseteq \relcomp S {Y \cap Z}$
By De Morgan's Laws: Relative Complement of Intersection:
- $X \subseteq \relcomp S Y \cup \relcomp S Z$
By Intersection with Subset is Subset:
- $X = \paren {\relcomp S Y \cup \relcomp S Z} \cap X$
By Intersection Distributes over Union:
- $X = \paren {\relcomp S Y \cap X} \cup \paren {\relcomp S Z \cap X}$
By definition of closed set and Relative Complement of Relative Complement:
- $X$, $\relcomp S Y$, and $\relcomp S Z$ are closed sets.
By Intersection of Closed Sets is Closed in Topological Space:
- $\relcomp S Y \cap X$, $\relcomp S Z \cap X$ are closed sets.
By definition of irreducible:
- $\relcomp S Y \cap X = X$ or $\relcomp S Z \cap X = X$
By Intersection with Subset is Subset:
- $X \subseteq \relcomp S Y$ or $X \subseteq \relcomp S Z$
By Relative Complement inverts Subsets and Relative Complement of Relative Complement:
- $Y \subseteq \relcomp S X$ or $Z \subseteq \relcomp S X$
Thus by definition of inclusion ordered set:
- $Y \preceq \relcomp S X$ or $Z \preceq \relcomp S X$
$\blacksquare$
Sources
- 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices
- Mizar article WAYBEL14:17