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!!!