Definition:Meet-Irreducible Open Set

Definition

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

Let $W \in \tau$.

Then $W$ is a meet-irreducible open set if and only if:

$W$ is meet-irreducible in the frame $\struct {\tau, \subseteq}$

That is, $W$ is a meet-irreducible open set if and only if:

$\forall U, V \in \tau : \paren {U \cap V \subseteq W \implies U \subseteq W \text { or } V \subseteq W}$

Also see

• Complement of Singleton Closure is Meet-Irreducible

• Results about meet-irreducible open sets can be found here.

Sources

• 2012: Jorge Picado and Aleš Pultr: Frames and Locales: Chapter $1$: Spaces and Lattices of Open Sets, $\S 1$ Sober spaces, Definition $1.1$