topological sum

Given two topological spaces $X$ and $Y$ , their topological sum is defined to be the set $X \coprod Y$ (see the entry disjoint union) equipped with the finest topology such that the inclusion maps from $X$ and $Y$ into $X \coprod Y$ are continuous. A basis for this topology consists of the union of the set of open subsets of $X$ and the set of open subsets of $Y$ .