Initial topology

The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.


The dual notion is the final topology, which for a given family of functions mapping to a set ${\displaystyle X}$ is the finest topology on ${\displaystyle X}$ that makes those functions continuous.

Definition[edit]

Given a set ${\displaystyle X}$ and an indexed family ${\displaystyle \left(Y_{i}\right)_{i\in I}}$ of topological spaces with functions $${\displaystyle f_{i}:X\to Y_{i},}$$ the initial topology ${\displaystyle \tau }$ on ${\displaystyle X}$ is the coarsest topology on ${\displaystyle X}$ such that each $${\displaystyle f_{i}:(X,\tau )\to Y_{i}}$$ is continuous.


Definition in terms of open sets


If ${\displaystyle \left(\tau _{i}\right)_{i\in I}}$ is a family of topologies ${\displaystyle X}$ indexed by ${\displaystyle I\neq \varnothing ,}$ then the least upper bound topology of these topologies is the coarsest topology on ${\displaystyle X}$ that is finer than each ${\displaystyle \tau _{i}.}$ This topology always exists and it is equal to the topology generated by ${\displaystyle {\textstyle \bigcup \limits _{i\in I}\tau _{i}}.}$^[3]


If for every ${\displaystyle i\in I,}$ ${\displaystyle \sigma _{i}}$ denotes the topology on ${\displaystyle Y_{i},}$ then ${\displaystyle f_{i}^{-1}\left(\sigma _{i}\right)=\left\{f_{i}^{-1}(V):V\in \sigma _{i}\right\}}$ is a topology on ${\displaystyle X}$, and the initial topology of the ${\displaystyle Y_{i}}$ by the mappings ${\displaystyle f_{i}}$ is the least upper bound topology of the ${\displaystyle I}$-indexed family of topologies ${\displaystyle f_{i}^{-1}\left(\sigma _{i}\right)}$ (for ${\displaystyle i\in I}$).^[3] Explicitly, the initial topology is the collection of open sets generated by all sets of the form ${\displaystyle f_{i}^{-1}(U),}$ where ${\displaystyle U}$ is an open set in ${\displaystyle Y_{i}}$ for some ${\displaystyle i\in I,}$ under finite intersections and arbitrary unions.


Sets of the form ${\displaystyle f_{i}^{-1}(V)}$ are often called cylinder sets. If ${\displaystyle I}$ contains exactly one element, then all the open sets of the initial topology ${\displaystyle (X,\tau )}$ are cylinder sets.