Element is Meet Irreducible iff Complement of Element is Irreducible

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $P = \left({\tau, \preceq}\right)$ be an ordered set where $\mathord\preceq = \mathord\subseteq \cap \left({\tau \times \tau}\right)$

Let $A \in \tau$ such that
 * $A \ne \top_P$

where $\top_P$ denotes the greatest element in $P$.

Then $A$ is meet irreducible in $P$ $\complement_S\left({A}\right)$ is irreducible

where $\complement_S\left({A}\right)$ denotes the relative complement of $A$ relative to $S$.