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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.

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up vote 2 down vote accepted

Unless I’m not awake yet, this is true whether or not $X$ is $T_0$.

Suppose, as you did, that $U\cap\operatorname{cl}\{x\}\ne\varnothing\ne V\cap\operatorname{cl}\{x\}$, and fix $u=U\cap\operatorname{cl}\{x\}$ and $v\in V\cap\operatorname{cl}\{x\}$. Then every open nbhd of $u$ contains $x$, so in particular $x\in U$. Similarly, $x\in V$. But then $x\in U\cap V$.

The point is that if $U$ is any open set in $X$, then $U\cap\operatorname{cl}\{x\}=\varnothing$ iff $x\notin U$.

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Thank you for your help! I think I must have been a little confused. Reading that part of the paper over and over again, it made me realize I need T$_{0}$ to prove a uniqueness, but I don't need that property to prove meet-irreducibility. Again, thanks for your help! – KarenVO Mar 16 '12 at 16:21

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