----- > [!proposition] Proposition. ([[topological boundary is idempotent iff boundary is empty interior]]) > Let $(X, \tau)$ be a [[topological space]] and $S \subset X$ a subset. Then $\text{Bd }(\text{Bd }S)=\text{Bd }S$ if and only if the [[boundary]] of $S$ has empty [[topological interior|interior]]. > [!proof]- Proof. ([[topological boundary is idempotent iff boundary is empty interior]]) > ~ > $\to$. Suppose $\text{Bd }(\text{Bd }S)=\text{Bd }S$. Recalling that $\text{Bd }S=\overline{S}-\text{Int }S$, we have $\underbrace{\overline{\overline{S}- \text{Int }S} - {\text{Int }\text{Bd }S}}_{\text{bd = closure - interior}}=\text{Bd }S,$ but since $\overline{S}-\text{Int }S$ is closed (boundaries are always closed) we can drop the bar on it, but then we're just left with $\text{Bd }S - \text{Int }\text{Bd }S=\text{Bd }S,$ enforcing that $\text{Int }\text{Bd }S=0$. > $\leftarrow.$ Suppose $\text{Bd }S$ has empty interior. Recall ([[closure is interior together with boundary|boundary is closure minus interior]]) that $\text{Bd Bd }S=\overline{\text{Bd }S}-\text{Int } \text{Bd }S$. We've assumed the $\text{Int }\text{Bd }S=\emptyset$, so this leaves us with $\text{Bd }\text{Bd }S = \overline{\text{Bd }S}=\text{Bd }S$ (remember that boundaries are closed as intersections of [[closed set|closed sets]]). ----- #### ---- #### 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 > ```