----- > [!proposition] Proposition. ([[continuity preserves connectedness]]) > The image of a [[connected]] [[topological space|space]] under a [[continuous]] map is [[connected]]. > [!proof]- Proof. ([[continuity preserves connectedness]]) Let $X,Y$ be [[topological space]]s with $X$ connected; let $f:X \to Y$ be [[connected]]. Implicitly we will use the fact that [[range restriction and expansion of continuous function is continuous]]. > We show the contrapositive. Suppose $f(X)$ is not [[connected]]; write $f(X)=U \sqcup V$ for nonempty $U$, $V$ open in $Y$. Then $X=f^{-1}(U \sqcup V)=f^{-1}(U) \sqcup f^{-1}(V),$ which is a disjoint union of nonempty open (by [[continuous|continuity]]) subsets of $X$ unioning to $X$ — i.e., a [[separation of a topological space|separation]] of $X$. Hence $X$ is not [[connected]]. ----- #### ---- #### 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 > ```