---- > [!definition] Definition. (Neighborhood basis) > A [[topological space]] $X$ is said to have a **neighborhood basis at $x$** if there is a collection $\mathscr{B}$ of [[neighborhood|open neighborhoods]] of $x$ s.t. each neighborhood of $x$ contains at least one of the elements of $\mathscr{B}$. > \ That is, $\mathscr{B}$ [[nestles in]] the open [[neighborhood|neighborhoods]] of $x$. ^definition > [!definition] Definition. ([[first-countable space]]) > A space that has a [[countably infinite|countable]] neighborhood basis at each point is said to be **first-countable**. > [!note] Remark. > First countable spaces are important because they allow the use of [[sequence|sequences]] to detect many topological properties. For example, see [[the sequence lemma]] and [[the sequential continuity lemma]]. ^note > [!basicexample] > - Every [[metrizable]] space is first-countable. > - Every [[second-countable space]] is first-countable. > [!proof] Proofs of Examples. > - [[TODO]] > - Obvious: if $\mathscr{B}$ is a countable [[topology generated by a basis|basis]] for the entire topology of $X$, then the subcollection of $\mathscr{B}$ of elements containing $x \in X$ is too. ---- #### ---- #### 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 > ```