The Stacks project

5.27 Miscellany

The following lemma applies to the underlying topological space associated to a quasi-separated scheme.

Lemma 5.27.1. Let $X$ be a topological space which

  1. has a basis of the topology consisting of quasi-compact opens, and

  2. has the property that the intersection of any two quasi-compact opens is quasi-compact.

Then

  1. $X$ is locally quasi-compact,

  2. a quasi-compact open $U \subset X$ is retrocompact,

  3. any quasi-compact open $U \subset X$ has a cofinal system of open coverings $\mathcal{U} : U = \bigcup _{j\in J} U_ j$ with $J$ finite and all $U_ j$ and $U_ j \cap U_{j'}$ quasi-compact,

  4. add more here.

Proof. Omitted. $\square$

Definition 5.27.2. Let $X$ be a topological space. We say $x \in X$ is an isolated point of $X$ if $\{ x\} $ is open in $X$.

Comments (3)

Comment #4068 by Harry Gindi on

I was informed today that it is well-known terminology that an abstract topological space is called quasiseparated if it satisfies condition 2 (quasicompact opens are closed under finite intersections). I looked for a definition here yesterday, and I did not find it. Could this definition also be added to the miscellany?

Cheers,

Harry

Comment #4069 by Harry Gindi on

I was informed today that it is well-known terminology that an abstract topological space is called quasiseparated if it satisfies condition 2 (quasicompact opens are closed under finite intersections). I looked for a definition here yesterday, and I did not find it. Could this definition also be added to the miscellany?

Cheers,

Harry

Comment #4145 by on

Yeah, hmm... as you will see I have everywhere where this happens in topology (both usual topology and on sites etc) just written out the condition in full... So it would be a rather large thing to make the change...

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