> [!definitionb] Definition > > Let $(X, \topo)$ be a topological space, $A \subset X$, $x \in X$ and $\cm(x)$ be a [[Neighbourhood Base|neighbourhood base]] at $x$. Then the following are equivalent: > 1. $x \in \bigcap_{B \supset X, B \text{ closed}}B$ is in every [[Closed Set|closed set]] containing $A$. > 2. For every [[Neighbourhood|neighbourhood]] $U \in \cn(x)$, $U \cap A \ne \emptyset$. > 3. For every $U \in \cm(x)$, $U \cap A \ne \emptyset$. > 4. There exists a [[Filter|filter]] $\fF \subset \pow{A}$ such that $x$ is a limit point of $\fF$. > 5. There exists a [[Net|net]] $\angles{x_\alpha}_{\alpha \in A} \subset A$ such that $x_\alpha \to x$. > 6. *(If $x$ has a countable neighbourhood base)* There exists a [[Sequence|sequence]] $\seq{x_j}$ that [[Limit|converges]] to $x$. > > If the above holds, then $x$ is an **adherent point** of $A$. The set of all such $x$ is the **topological closure** of $A$. > > *Proof*. $(1) \Leftrightarrow (2)$: $x \not\in \overline{A}$ if and only if there exists $U \in \cn(x)^o$ such that $U \cap A \ne \emptyset$. > > $(2) \Rightarrow (3)$: $\cn(x) \supset \cm(x)$. > > $(3) \Rightarrow (4)$: Let $\fF = \bracs{U \cap A: U \in \cm(x)}$, then $\fF$ contains all relative neighbourhoods of $x$ in $A$, and converges to $x$. > > $(4) \Rightarrow (5)$: $\fF$ is directed under reverse inclusion. For every $U \in \fF$, let $x_U \in U$, then $x_U \to x$ and $\angles{x_U}_{U \in \fF}$ is a desired net. > > $(5) \Rightarrow (2)$: Since $x_\alpha \to x$ and $\net{x} \subset A$, every neighbourhood of $x$ intersects $A$. > [!definition] > > Let $A \subset X$, and $(Y \subset X, \topo_{Y})$ be an [[Relative Topology|induced topology]]. The topological closure of $A$ in $Y$, $\overline{A}_{Y}$ is > $ > \overline{A}_{Y} = \overline{A} \cap X > $