Convergence

Definition: Limit points

Let $X$ be a space and set $A\subseteq X$ A point $p\in X$ is a limit point of A if every neighborhood of $p$ contains a point of $A$ other than $p$ OR _accumulation point

$A^{\prime }$ denotes the set of limit points.

For any topology space we have $\bar{A}=A\cup A^{\prime }$ Topology by James Munkres, p.99 $⟹$ $A^{\prime }\subseteq \bar{A}$

NOTE

In any topological space, if a subset $A$ has no isolated points, then its set of limit points is exactly equal to its closure.

In this case, $p$ and $b$ are limit points

Definition: Isolated points

A point $p\in A$ is an isolated point of A if $p$ has a neighborhood $U$ in $X$ with $U\cap A=\{p\}$ In this case, $a$ is a isolated point

Definition: dense

A subset $A$ of $X$ is dense if $\bar{A}=X$ A subset $A$ is dense in $X$ $⟺$ every non-empty open $B\subseteq X$ contains a point of $A$

Example

$\mathbb{Q}$ is dense in $\mathbb{R}$

Definition: Convergence

$(x_i{)}_{i=1}^{\infty }$ is a sequence of points in $X$, and $x\in X$, we say the sequence converges to the limit $x$, if for every neighborhood $U$ of $x$ there is $N\in \mathbb{N}$ such that $x_i\in U$ for all $i\ge N$

we also say $x$ is a limit point of this sequence

In other word

For any neighborhood $U$ of $x$, there exists all the elements in the sequence from a certain index.

similar to definition of convergence in Euclidean space In ${\mathbb{R}}^2$, we take a ball of $x$ instead of an arbitrary neighborhood

If $x_i\to x$ then $x\in \bar{A}$ (limits lies in closure)