Initial topology

In <a href="/facts/General_topology/ezC7B0yP">general topology</a> and related areas of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the initial topology (or induced topology or strong topology or limit topology or projective topology) on a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 with respect to a family of functions on 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 is the <a href="/facts/Coarsest_topology/5Q9F2fjf">coarsest topology</a> on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 that makes those functions <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a>.
The <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a> and <a href="/facts/Product_topology/0hg02IaX">product topology</a> constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these.
The <a href="/facts/Duality_(mathematics)/8jGuyQIU">dual</a> notion is the <a href="/facts/Final_topology/gk2UV5va">final topology</a>, which for a given family of functions mapping to a set 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is the <a href="/facts/Finest_topology/5Q9F2fjf">finest topology</a> on 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 that makes those functions continuous.

Initial topology open-in-new

Initial topology