TITLE: Constructible subset of constructible set
QUESTION [6 upvotes]: Let $X$ be a topological space. Let $F \subset E \subset X$ be subsets. Assume that $E$ is constructible in $X$ and that $F$ is constructible in $E$. Is it true that $F$ is constructible in $X$?
We use the definition of constructible as defined in EGA I (first edition).
This is true if $X$ has a basis for the topology consisting of quasi-compact opens, see Lemma Tag 09YJ. In particular, it holds if $X$ is a scheme. Thus our question is a bit frivolous as the important use case is covered.
Still: Is there a counter example for general $X$? Is it true if $X$ is quasi-compact?

REPLY [3 votes]: If $F$ is open in $E$ (where $E$ carries the subspace topology from $X$), then $F = E \cap U$ where $U$ is open in $X$, hence $F$ is constructible in $X$ (since constructibles are closed under intersection). Similarly, if $F$ is closed in $E$, then $F$ is constructible in $X$. 
The constructibles in $E$ constitute the smallest Boolean algebra that contains the open and closed sets in $E$. The constructibles in $X$ are a Boolean algebra which, we have just seen, include the open and closed sets in $E$. Hence constructibles in $E$ are included in the the constructibles of $X$, as anticipated.