Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer

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 $x\notin U$.

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.