Way Below if Between is Compact Set in Ordered Set of Topology
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $L = \struct {\tau, \preceq}$ be an ordered set where $\preceq \mathop = \subseteq\restriction_{\tau \times \tau}$
Let $x, y \in \tau$ such that:
- $\exists H \subseteq S: x \subseteq H \subseteq y \land H$ is compact
Then:
- $x \ll y$
Proof
Let $D$ be a directed subset of $\tau$ such that:
- $y \preceq \sup D$
By proof of Topology forms Complete Lattice:
- $\ds y \subseteq \bigcup D$
By Subset Relation is Transitive:
- $\ds H \subseteq \bigcup D$
By definition:
- $D$ is open cover of $H$
By definition of compact:
- $H$ has finite subcover $\GG$ of $D$
By Directed iff Finite Subsets have Upper Bounds:
- $\exists d \in D: d$ is upper bound for $\GG$
By definitions of upper bound and of $\preceq$:
- $\forall z \in \GG: z \subseteq d$
By Union is Smallest Superset/Set of Sets:
- $\ds \bigcup \GG \subseteq d$
By definition of cover:
- $\ds H \subseteq \bigcup \GG$
By Subset Relation is Transitive:
- $x \subseteq d$
Thus by definition of $\preceq$:
- $x \preceq d$
Thus by definition of way below relation:
- $x \ll y$
$\blacksquare$
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 WAYBEL_3:38