---- > [!definition] Definition. ([[balanced category]]) > We call a [[category]] $\mathsf{C}$ **balanced** if [[monomorphism]] + [[epimorphism]] implies [[isomorphism]] in $\mathsf{C}$. ^definition > [!basicnonexample] > While it is true that, in $\mathsf{Top}$, [[monomorphism]] $\iff$ [[injection]] and [[epimorphism]] $\iff$ [[surjection]] hold, $\mathsf{Top}$ is still not balanced since the inverse of a [[bijection|bijective]] [[continuous]] map need not be [[continuous]]. ^nonexample ---- #### ---- #### 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 > ```