--- > [!definition]+ Definition. ([[disjoint union topology]]) > Consider the [[disjoint union|disjoint set-union]] $X:=\bigsqcup_{i \in I}X_{i}$ of the family $\{ X_{i} : i \in I \}$ of [[topological space|topological spaces]] indexed by $I$. For each $i \in I$, let $\varphi_{i}: X_{i} \to X, x \xmapsto{\varphi_{i}} (x,i)$ be the canonical [[injection]]. The **disjoint union topology on $X$** is defined by declaring a subset $U$ of $X$ to be open if and only if $\varphi_{i} ^{-1}(U)$ is open in $X_{i}$ for each $i \in I$. ^definition --- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch03