topology induced by some family of maps valued in . Then the final topology on induced by is equal to the final topology on induced by the maps As a consequence: if is the final topology on induced by the family and if is any surjective map valued in some topological space then is a quotient map if and only if has the final topology induced by the maps By the universal property of the disjoint union topology we know that given any family of continuous maps there is a unique continuous map that is compatible with the natural injections. If the family of maps (i.e. each lies in the image of some ) then the map will be a quotient map if and only if has the final topology induced by the maps Effects of changing the family of maps Throughout, let be a family of -valued maps with each map being of the form and let denote the final topology on induced by The definition of the final topology guarantees that for every index the map is continuous. For any subset the final topology on will be than (and possibly equal to) the topology ; that is, implies where set equality might hold even if is a proper subset of If is any topology on such that and is continuous for every index then must be than (meaning that and this will be written ) and moreover, for any subset the topology will also be than the final topology that induces on (because ); that is, Suppose that in addition, is an -indexed family of -valued maps whose domains are topological spaces If every is continuous then adding these maps to the family will change the final topology on that is, Explicitly, this means that the final topology on induced by the "extended family" is equal to the final topology induced by the original family However, had there instead existed even just one map such that was continuous, then the final topology on induced by the "extended family" would necessarily be than the final topology induced by that is, (see this footnote for an explanation). Coherence with subspaces Let be a topological space and let be a family of subspaces of where importantly, the word "sub" is used to indicate that each subset is endowed with the subspace topology inherited from The space is said to be with