---- > [!definition] Definition. ([[connected]]) > A [[topological space]] $X$ is said to be **connected** if there does not exist a [[separation of a topological space|separation]] of $X$. > > [!equivalence] Equivalences. > - [[clopen characterization of connectedness]] > - [[De Morgan's Laws|De Morgan's (second) law]] hints at a characterization of connectedness in terms of [[closed set|closed sets]]: $X$ is disconnected if and only if there exist closed $C,D \subset X$ satisfying $C \cup D=X$ and $C \cap D=\emptyset$,[^1], that is $C \sqcup D=X$. [^1]: Omitted above is the observation that the condition $(X-C) \sqcup (X-D)=X$ is equivalent to the condition $C \cup D=X$. Indeed, $a \in (X-C) \sqcup (X-D)$ iff $a \notin C$ *xor* $a \notin D$, or equivalently, iff $a \in C$ or $a \in D$ > [!basicproperties] Properties. > - [[connected subspace of separated set lies in one constituent]] > - [[continuity preserves connectedness]] > - [[adjoining limit points preserves connectedness]] > - [[adjoining limit points preserves connectedness|closure preserves connectedness]] > - [[arbitrary union of nontrivially-intersecting connected subspaces is connected]] > - [[coarsening preserves connectedness]] ---- ---- #### 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 > ```