---- > [!definition] Definition. ([[converge]]) > In an arbitrary [[topological space]] $X$, one says that a [[sequence]] $x_{1}, x_{2},\dots$ of points of the space $X$ **converges** to the point $x \in X$ provided that for each [[neighborhood]] $U$ of $x$, there exists $N \in \mathbb{N}$ such that $x_{n} \in U$ for all $n \geq N$. We call $x$ a **limit** of the sequence. > [!equivalence] > If $\mathscr{B}$ is a [[basis for a topology|basis]] for $\tau_{X}$, then it is enough to check that for every [[basis for a topology|basis element]] $B$ containing $x$, there exists $N \in \mathbb{N}$ s.t. $x_{n} \in B$ for all $n \geq N$. ^711ef2 > [!justification] Proof of Equivalence. > Suppose for every basis element $B \in \mathscr{B}$ containing $x$, there exists $N \in \mathbb{N}$ s.t. $x_{n} \in B$ for all $n \geq N$. Let $U$ be an arbitrary [[neighborhood]] of $x$. By [[open sets are unions of basis elements]] we know that we can write $U=\bigcup_{\alpha}^{}B_{\alpha}$ for some indexed collection of basis elements $\{ B_{\alpha} \}_{\alpha}$. The point $x$ lies in one of these basis elements, call it $B$, and using this we can fix $N \in \mathbb{N}$ s.t. for all $n \geq N$ we have $x_{n} \in B$. Then, it follows that for all $n \geq N$ we have $x_{n} \in B \subset U$ and in particular $x_{n} \in U$, as required. ---- #### ---- #### 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 > ``` # In a [[topological space]] a [[sequence]] $(x_n) \to x_0$ if for every [[open set]] $\Omega$ that contains $x_0$ almost every term in the [[sequence]] (all but [[finite]]ly many terms) are in $\Omega$. #notFormatted