nLab zero-set structure



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



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

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

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

  • If Z 1,Z 2𝒵Z_1, Z_2 \in \mathcal{Z} and Z 1Z 2=Z_1 \cap Z_2 = \emptyset, then there are V 1,V 2𝒵V_1, V_2 \in \mathcal{Z} with Z 1V 1 cZ_1 \subseteq V_1^c, Z 2V 2 cZ_2 \subseteq V_2^c, and V 1 cV 2 c=V_1^c \cap V_2^c = \emptyset.

See also


  • 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)

Created on May 24, 2023 at 21:36:14. See the history of this page for a list of all contributions to it.