Metric space: Open and closed Balls, neighbourhood, open and closed set - Lecture II 

The distance function provides an idea of the "surrounding" of a point. Given a point $a$ and a number $r>0$, to distinguish between those points near $a$, then $d(x,a)<r$ must be satisfied. This brings us to the concept of open balls. In addition we will talk about the neighbourhood of a set, limit points, interior point and boundary points. These are special properties that make the study of "distance" more interesting.

A summary of balls.


So what is a ball in metric space?
Generally, in mathematics a ball is simply defined as the volume space bounded by a sphere. There are three types of balls, the open ball, this is a ball that excludes all its boundary point, while the closed ball includes all its boundary points and the sphere which is ball that also includes all boundary points making it very similar to the closed ball with only few distinction.
Let $(X,d)$ be a metric space, Given a ball with a point $a\in{X}$ which is the center of the ball, and $B_r(a)$ radius of the ball denoted by a real number $r>0$. The sets:
  1.  $B_r(a):=\{x\in{X}:d(x,a)<r\}$ is an open ball. 
  2.  $B_r(a):=\{x\in{X}:d(x,a)\leq{r}\}$ is a closed ball.
  3.  $S_r(a):=\{x\in{X}:d(x,a)=r\}$ is a sphere.

From the above mathematical expressions, an open ball of radius $r$ is the set of all points in $X$ whose distance from the center of the ball is less than $r$.
Also the closed  ball of radius $r$ is the set of all points in $X$
whose distance from the center of the ball is less than or equal to $r$ and the sphere of radius $r$ is the set of all points in $X$ whose distance from the center of the ball is equal to $r$. Now, let's take a look at other beautiful concepts of balls. 
A point $x$ of a set $A$ is called an interior point of $A$ when it can be “surrounded
completely” by points of $A$, where $A$ is a subset of the set $X$, this is defined as: for $r>0$, $B_r(a)\subseteq{A}$. This also defines the neighbourhood of the set $A$, so we say that $A$ is a neighborhood of the point $x$. 
The point $x$ that is not in $A$ is said to be an exterior point of $A$ if $r>0$ and $B_r(x)\subseteq{X\A}$. In this case, all other points in $A$ are called boundary points of $A$. 
The union of the interior points and boundary points is called the closure of $A$ this is defined by, let $Int(A)$ be the set of interior points in $A$, $b(A)$ be the set of boundary points in $A$ and $cl(A)$ be the set of closure points in $A$ then $cl(A):=int(A)\bigcup{b(A)}$. 
Since $A$ is a subset of the set $X$, then we say the set $A$ is an open set in $X$ if all its points are interior points.
While the set $B$ is said to be a closed set in $X$ if $X\B$ is open in $X$, remember $B$ must be a subset of $X$. 
A point $b$ (not necessarily in $A$) is a limit point of a set $A$ when every ball
around it contains other points of $A$, i.e for $\epsilon>0$ there exists $a\neq{b}$ such that $a\in{A}\bigcap{B}_\epsilon(b)$