Polytopological space

An ${\displaystyle L}$‑topological space ${\displaystyle (X,\tau )}$ is a set ${\displaystyle X}$ together with a monotone map ${\displaystyle \tau :L\to }$ Top${\displaystyle (X)}$ where ${\displaystyle (L,\leq )}$ is a partially ordered set and Top${\displaystyle (X)}$ is the set of all possible topologies on ${\displaystyle X,}$ ordered by inclusion. When the partial order ${\displaystyle \leq }$ is a linear order, then ${\displaystyle (X,\tau )}$ is called a polytopological space. Taking ${\displaystyle L}$ to be the ordinal number ${\displaystyle n=\{0,1,\dots ,n-1\},}$ an ${\displaystyle n}$‑topological space ${\displaystyle (X,\tau _{0},\dots ,\tau _{n-1})}$ can be thought of as a set ${\displaystyle X}$ together with ${\displaystyle n}$ topologies ${\displaystyle \tau _{0}\subseteq \dots \subseteq \tau _{n-1}}$ on it (or ${\displaystyle \tau _{0}\supseteq \dots \supseteq \tau _{n-1},}$ depending on preference). More generally, a multitopological space ${\displaystyle (X,\tau )}$ is a set ${\displaystyle X}$ together with an arbitrary family ${\displaystyle \tau }$ of topologies on ${\displaystyle X.}$[2]

• Bitopological space