Moniker1998
Moniker1998
A space with this property is called $\sigma$-connected, see Continuum theory by Nadler, chapter 5 section 4 Every continuum admits the stronger property of being $\sigma$-connected, a theorem by Sierpiński
I've got a response, K.P. Hart gave a very simple proof of this using Shirota's theorem [See here](https://mathoverflow.net/a/469610/150060) I think this answer would be great for a reference. Shirota's theorem...
Seems like weak local compactness is closed under disjoint union (wikipedia provides example of disjoint union of one-point compactification of Q and a particular point topology on infinite set)/ So...
Nonetheless, maybe it would be nice to have a connected example on pi-base instead.
S42 Right Ray Topology on the Reals 1. Weakly locally compact: If x is a point then [x-1, inf) is a compact neighbourhood of x 2. Not locally relatively compact:...
[S46](https://github.com/pi-base/data/issues/607) Interlocking interval topology seems to be Alexandrov too
My opinion is that I'm perfectly fine with not having the fourth definition of locally compact space, since as you say it can be described as weakly locally compact regular...
I wrote a detailed answer for why [this space is not realcompact here](https://math.stackexchange.com/a/4726056/476484).
I also agree that Willard's definition looks nice, but when actually trying to prove that something is realcompact, it's mostly useless. I think that's why it's not as popular as...