--- > [!proposition]+ Proposition. ([[adjoining limit points preserves connectedness]]) > Let $A$ be a [[connected|connected]] [[subspace]] of $X$. If $A \subset B \subset \overline{A}$, then $B$ is also [[connected]]. > > Put differently: If $B$ is formed by adjoining to the [[connected|connected]] [[subspace topology|subspace]] $A$ some or all of its [[limit point|limit points]], then $B$ is [[connected|connected]]. > > A special case is of course $B=\overline{A}$: the [[closure]] of a [[connected|connected]] set is [[connected|connected]]. ^proposition > [!proof]+ Proof. ([[adjoining limit points preserves connectedness]]) > > The result is trivial if $A=B$, so suppose $B-A \neq \emptyset$. > > Suppose $B$ is not connected, $B=C \sqcup D$ for some nonempty $C,D$ open in $B$. By [[connected subspace of separated set lies in one constituent]], we have $A \subset C$ or $A \subset D$, WLOG $A \subset C$. Then $\overline{A} \subset \overline{C}$ ([[closure and union|see here]]) but $B \subset \overline{A}$, and this contradictions the assumption that $D$ was not empty. ^proof --- #### --- #### 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 > ```