nLab zero-set structure

 Definition

Given a set $X$, a zero-set structure on $X$ is a $\sigma$-topology $\mathcal{Z}$ such that

• For every pair of distinct points in $X$ there is an open set $Z \in \mathcal{Z}$ containing precisely one of these points.

• If $Z \in \mathcal{Z}$ then there are $Z_n \in \mathcal{Z}$ for all natural numbers $n$ such that $Z^c = \bigcup_{n \in \mathbb{N}} Z_n$.

• If $Z_1, Z_2 \in \mathcal{Z}$ and $Z_1 \cap Z_2 = \emptyset$, then there are $V_1, V_2 \in \mathcal{Z}$ with $Z_1 \subseteq V_1^c$, $Z_2 \subseteq V_2^c$, and $V_1^c \cap V_2^c = \emptyset$.

See also

• zero set
• sigma-topology

References

• Fedor Petrov, “countable” topology, MathOverflow (web)

• Hugh Gordon, Rings of functions determined by zero-sets. Hugh Gordon. Pacific J. Math. Volume 36, Number 1 (1971), 133-157. (pdf)