Is every closed subset of quasi-separated space quasi-separated?

A topological space is called quasi-separated if the intersection of two quasi-compact opens is quasi-compact. Here quasi-compact means every open cover has a finite subcover.


1 Answer 1


Let $X$ be a space with the following property: each point has a n.h. basis consisting of quasi-compact opens.

Then (since any closed subset of a quasi-compact set is quasi-compact), we deduce that any closed subset $Z$ of $X$ inherits this property.

Furthermore, one easily sees that any quasi-compact open subset $V$ of $Z$ may be written in the form $V = U \cap Z,$ where $U$ is a quasi-compact open subset of $X$. (Without making our initial assumption on $X$, this property need not hold; e.g. consider a compact subset $Z$ of $X = \mathbb R^n$.)

Suppose now that $X$ is also quasi-separated, and let $Z$ be a closed subset of $X$. If $V_1$ and $V_2$ are two quasi-compact open subsets of $Z$, we may write $V_i = U_i \cap Z$, where $U_i$ is quasi-compact open in $X$. Then $U_1 \cap U_2$ is quasi-compact by assumption, and so $V_1 \cap V_2 = U_1 \cap U_2 \cap Z$ is also quasi-compact; hence $Z$ is quasi-separated.

(I'm not sure what's true if we don't make the additional assumption on $X$; have you looked in the topology section of the Stacks Project?)


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