----- > [!proposition] Proposition. ([[clopen characterization of connectedness]]) > A [[topological space]] $X$ is [[connected]] if and only if the only [[clopen set|clopen]] subsets of $X$ are $\emptyset$ and $X$ itself. > [!note] Remark. > This result can be used in 'proofs by continuous induction'. The idea is that one wants to show property $\mathrm{P}$ holds for all $x \in X$. To do this, one defines $A:=\{ x \in X : \text{ property P holds} \} \subset X$. They then check the "base case" that $A \neq \emptyset$. Then they check that $A$ is clopen in $X$, which conceptually is the statement "if property P holds at one point it also holds at points nearby" ^note > [!proof]- Proof. ([[clopen characterization of connectedness]]) > ~ > Suppose $U \subset X$ is [[clopen set|clopen]] in $X$, but $U \neq X$ and $U \neq \emptyset$. Then $X-U$ is [[clopen set|clopen]] too. But $U \cap (X-U) = \emptyset$ and $U \cup (X-U)=X$, meaning that $U$ and $X-U$ form a [[separation of a topological space|separation]] of $X$ and thus $X$ is not [[connected]]. > Conversely, suppose $X$ is not [[connected]]. Then $X=U \sqcup V$ for nonempty, nonuniverse open $U,V \subset X$. Since $U \cap V = \emptyset$, we have that $X-U=V$ and $X-V=U$. But the [[closed sets behave complementarily to open sets|complement of]] of an [[open set]] is a [[closed set]], meaning that we have just exhibited than $V$ and $U$ are both [[closed set|closed]]. ----- #### ---- #### 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 > ```