I'm reading the paper 'Frames' by A. Pultr and I'm having trouble proving the following:
Let $\Omega(X)$ be the system of all open sets of X. This is a complete lattice. Let now X be a T$_{0}$ space, then $X \setminus \overline{\{x\}}$ is meet-irreducible, i.e. $\forall U,V \in \Omega(X)$ $U \cap V \subseteq X \setminus \overline{\{x\}} \Rightarrow U \subseteq X \setminus \overline{\{x\}} \ or \ V \subseteq X \setminus \overline{\{x\}}.$
I tried the following (proof by contraposition): $U \nsubseteq X \setminus \overline{\{x\}} \ and \ V \nsubseteq X \setminus \overline{\{x\}} \\ \Rightarrow U \cap \overline{\{x\}} \neq \emptyset \ and \ V \cap \overline{\{x\}} \neq \emptyset\\ \Rightarrow \exists u \in U: u \in \overline{ \{x\} } \ and \ \exists v \in V: v \in \overline{\{x\}}\\ \Rightarrow \exists u \in U: \forall A \in \mathcal{V}(u): x \in A \ and \ \exists v \in V: \forall B \in \mathcal{V}(v): x \in B \\ \Rightarrow \exists u \in U: \exists K \in \mathcal{V}(x): u \notin K \ and \ \exists v \in V: \forall L \in \mathcal{V}(x): v \notin L \ (becauce \ X \ is \ T_{0}) \\$
I need to find that $U \cap V \cap \overline{\{x\}} \neq \emptyset$, but i don't see how.
Any help would be appreciated.