Hausdorff Spaces are Sober

Our main reference is

📚 Jorge Picado & Aleš Pultr Frames and Locales: topology without points. Springer Basel, Series Frontiers in Mathematics, Vol. 28, xx + 400 pp., 2012 (ISBN: 978-3-0348-0153-9).

On page 2, there is a demonstration for the fact that

A Hausdorff space is always sober.

Being a little picky, I have found the demonstration on the reference book, although very easy… unnecessarily complicated. This is a very small post where I am going to demonstrate this fact in a much simpler way. It is basically the same, but simpler.

Let be an open set such that there are two distinct such that . Since is Hausdorff, there are two disjoint open neighbourhoods and of and . Of course, none is contained in . But . Therefore, is not meet-irreducible.

That is, every meet-irreducible must contain every point except for one. It is, therefore, of the shape for some .

🥳 🍻 😎

Cheers!!!


Published

Category

point-free topology

Tags