Properties:: [[closure is interior together with boundary]], [[empty boundary iff clopen]], [[open iff boundary is closure minus set]] Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* ---- - Let $X$ be a [[topological space]]. > [!definition] Definition. ([[boundary]]) > If $A \subset X$, we define the **boundary of $A$** by the equation $\text{Bd }A := \overline{A} \cap \overline{X-A}.$ > ---- #### ---- #### 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 > ``` # definition The [[boundary]] of a $A \subset \mathbb{R}^n$ is the set of those points in $\mathbb{R}^n$ which belong neither to $\text{Int } A$ nor $\text{Ext } A$, i.e., those points which belong to neither the [[topological interior]] of $A$ nor the [[exterior]] of $A$. It is denoted $\text{Bd } A$.