Final topology

Remarks

This family of sets is a topology, because preimages commute with unions and intersections. It is the finest topology on $Y$ for which every $f_i$ is continuous. Universal property: for a topological space $Z$, a map $g \colon Y \to Z$ is continuous for the final topology if and only if $g \circ f_i$ is continuous for every $i$.

Example: for a surjection $q \colon X \to Y$ from a space $X$, the final topology for $q$ is the quotient topology. Example: for the injections of the pieces into a disjoint union, the final topology is the disjoint union topology.

Non-example: let $q \colon \R \to \R$ be the identity, with the usual topology on the source. The trivial topology $\set{\varnothing, \R}$ on the target makes $q$ continuous but is not the final topology. The set $(0, 1)$ has an open preimage, yet it is not open in the trivial topology.

The final topology is also called the strong topology or the coinduced topology for the family.