math lessons - 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 $f_i: X \to Y_i$ be a family of functions, each from a fixed set $X$ to a topological space $Y i$ . The initial topology on $X$ (with respect to the family of functions) is the weakest topology such that every $f i$ is continuous. The topology is generated by sets of the form $f_i^{-1}(U)$ , where $U$ is an open set in $Y i$ .

Several topological constructions can be regarded as special cases of the initial topology. For example, the subspace topology for $X \subset Y$ 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.

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