---- > [!definition] Definition. ([[evenly covered]]) > Let $p: E \to B$ be a [[continuous]] [[surjection]] between [[topological space|topological spaces]]. The open set $U \subset B$ is said to be **evenly covered by $p$** if $p ^{-1}(U)$ can be written as the union of disjoint open sets $V_{\alpha}$ in $E$ such that for each $\alpha$, $p |_{V_{\alpha}}$ is a [[homeomorphism]] onto $U$. The collection $\{ V_{\alpha} \}$ is called a **partition of $p ^{-1}(U)$ into slices**. > \ > Note that if $U$ is evenly covered by $p$ and $W$ is an open set contained in $U$, then $W$ is also evenly covered by $p$. > [!note] Remark. > If $U$ is an open set that is evenly covered by $p$, we often picture the set $p ^{-1}(U)$ as a 'stack of pancakes', each having the same size and shape as $U$, floating in the air above $U$; the map $p$ squashes them all down onto $U$. > ![[CleanShot 2024-03-29 at [email protected]]] > ---- #### ---- #### 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 > ```