Quotient space of an algebraic stack

In algebraic geometry, the quotient space of an algebraic stack F, denoted by |F|, is a topological space which as a set is the set of all integral substacks of F and which then is given a "Zariski topology": an open subset has a form $$|U| \subset |F|$$ for some open substack U of F.

The construction $$X \mapsto |X|$$ is functorial; i.e., each morphism $$f: X \to Y$$ of algebraic stacks determines a continuous map $$f: |X| \to |Y|$$.

An algebraic stack X is punctual if $$|X|$$ is a point.

When X is a moduli stack, the quotient space $$|X|$$ is called the moduli space of X. If $$f: X \to Y$$ is a morphism of algebraic stacks that induces a homeomorphism $$f: |X| \overset{\sim}\to |Y|$$, then Y is called a coarse moduli stack of X. ("The" coarse moduli requires a universality.)