---- > [!definition] Definition. ([[neighborhood]]) > If $X$ is a [[topological space]] and $p \in X$, a **neighborhood** of $p$ is a subset $V \subset X$ that includes an [[open set]] $U$ containing $p$: $p \in U \subset V \subset X.$ > Equivalently, a [[neighborhood]] of $p$ is a subset $V \subset X$ whose [[topological interior|topological interior]] contains $p$. > [!attention] Warning. > Sometimes the term "neighborhood of $x_{0}quot; merely refers to an [[open set]] containing $x_{0}$. The above definition subsumes this one. However, because different classes I have taken prefer different versions of the definition, it will have to be inferred from context when 'neighborhood' actually means 'open neighborhood' in these notes (which is quite often). ---- #### ---- #### 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 > ```