Science Fair Project Encyclopedia
Initial topology
In mathematics, the initial topology on a set is the weakest topology that makes one or more specified functions on that set continuous.
Let 
be a family of functions, each from a fixed set X to a topological space Yi. The initial topology on X (with respect to the family of functions) is the weakest topology such that every fi is continuous. The topology is generated by sets of the form 
, where U is an open set in Yi.
Several topological constructions can be regarded as special cases of the initial topology. For example, the subspace topology for 
is the initial topology with respect to the inclusion map. The product topology is the initial topology with respect to the family of projection maps.
See also
- final topology
Last updated: 10-17-2005 04:51:55
10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details