Characterization of Prime Element in Inclusion Ordered Set of Topology

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $L = \struct {\tau, \preceq}$ be an inclusion ordered set of $\tau$.

Let $Z \in \tau$.

Then $Z$ is prime element in $L$ if and only if:

$\forall X, Y \in \tau: X \cap Y \subseteq Z \implies X \subseteq Z \lor Y \subseteq Z$

Proof

Sufficient Condition

Assume that

$Z$ is prime element in $L$.

Let $X, T \in \tau$ such that:

$X \cap Y \subseteq Z$

By Join and Meet in Inclusion Ordered Set of Topology and definition of inclusion ordered set:

$X \wedge Y \preceq Z$

By definition of prime element:

$X \preceq Z$ or $Y \preceq Z$

Thus by definition of inclusion ordered set:

$X \subseteq Z$ or $Y \subseteq Z$

$\Box$

Necessary Condition

Assume that: L$\forall X, Y \in \tau: X \cap Y \subseteq Z \implies X \subseteq Z \lor Y \subseteq Z$

Let $X, Y \in \tau$ such that:

$X \wedge Y \preceq Z$

By Join and Meet in Inclusion Ordered Set of Topology and definition of inclusion ordered set:

$X \cap Y \subseteq Z$

By assumption:

$X \subseteq Z$ or $Y \subseteq Z$

Thus by definition of inclusion ordered set:

$X \preceq Z$ or $Y \preceq Z$

$\blacksquare$

Sources

• Mizar article WAYBEL14:19