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1. • CommentRowNumber1.
   • CommentAuthorporton
   • CommentTimeOct 17th 2014
   • (edited Oct 17th 2014)
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   Author: porton
   Format: MarkdownItexWhy is definition 6 equivalent to definition 1 for frames in [compact space](http://ncatlab.org/nlab/show/compact+space)?

   Why is definition 6 equivalent to definition 1 for frames in compact space?

2. • CommentRowNumber2.
   • CommentAuthorporton
   • CommentTimeOct 17th 2014
   • (edited Oct 17th 2014)
   • PermaLink
   Author: porton
   Format: MarkdownItexI have found a counter-example to Definition 6 in [compact space](http://ncatlab.org/nlab/show/compact+space). Consider the set $Q$ of principal filters on the set $\mathbb{R}$ (opens of the frame of opens set in $\beta\mathbb{R}$) generated by sets $(-\epsilon;\epsilon)$ where $\epsilon$ passes through all positive reals. It's join on the set of filters the infinitely small neighborhood $\Delta$ of zero. But $Q\notin\Delta$. But the set of filters on reals is a compact frame ([see here](http://www.mathematics21.org/binaries/todd-notes.pdf)). A contradiction with definition 6. Most probably the counter-example is wrong and nLab is correct. But where is the error?

   I have found a counter-example to Definition 6 in compact space.

   Consider the set $Q$ of principal filters on the set $\mathbb{R}$ (opens of the frame of opens set in $\beta\mathbb{R}$) generated by sets $(-\epsilon;\epsilon)$ where $\epsilon$ passes through all positive reals.

   It’s join on the set of filters the infinitely small neighborhood $\Delta$ of zero. But $Q\notin\Delta$. But the set of filters on reals is a compact frame (see here). A contradiction with definition 6.

   Most probably the counter-example is wrong and nLab is correct. But where is the error?

3. • CommentRowNumber3.
   • CommentAuthorporton
   • CommentTimeOct 17th 2014
   • (edited Oct 17th 2014)
   • PermaLink
   Author: porton
   Format: MarkdownItexI've understood what is my error. Here $X=\mathbb{R}$ (but I thought that $X$ is an arbitrary subset of $\mathbb{R}$).

   I’ve understood what is my error. Here $X=\mathbb{R}$ (but I thought that $X$ is an arbitrary subset of $\mathbb{R}$).