In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X , {\displaystyle X,} with respect to a family of functions on X , {\displaystyle X,} is the coarsest topology on X {\displaystyle X} that makes those functions continuous. 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 Y {\displaystyle Y} is the finest topology on Y {\displaystyle Y} that makes those functions continuous.
Definition Given a set X {\displaystyle X} and an indexed family ( Y i ) i ∈ I {\displaystyle \left(Y_{i}\right)_{i\in I}} of topological spaces with functions
f i : X → Y i , {\displaystyle f_{i}:X\to Y_{i},}
the initial topology τ {\displaystyle \tau } on X {\displaystyle X} is the coarsest topology on X {\displaystyle X} such that each
f i : ( X , τ ) → Y i {\displaystyle f_{i}:(X,\tau )\to Y_{i}}
is continuous. Definition in terms of open sets If ( τ i ) i ∈ I {\displaystyle \left(\tau _{i}\right)_{i\in I}} is a family of topologies X {\displaystyle X} indexed by I ≠ ∅ , {\displaystyle I\neq \varnothing ,} then the least upper bound topology of these topologies is the coarsest topology on X {\displaystyle X} that is finer than each τ i . {\displaystyle \tau _{i}.} This topology always exists and it is equal to the topology generated by ⋃ i ∈ I τ i . {\textstyle \bigcup _{i\in I}\tau _{i}.}
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