Metric spaces

Definition 1(Definition).

A set $X$, whose elements are called points, is said to be a metric space if with any two points $p$ and $q$ of $X$ there is associated a real number $d(p,q)$, called the distance between $p$ and $q$, such that

• $d(p,q)>0$ if $p\ne q$
• $d(p,p)=0$
• $d(p,q)=d(q,p)$
• $d(p,q)\le d(p,r)+d(r,q)$
  The function $d$ is called a distance function, or a metric.

Observe:

• A metric space requires $\mathbb{R}$ to exist.
• Every subset $Y$ of a metric space $X$ is also a metric space.
• The Euclidean spaces ${\mathbb{R}}^k$ are metric spaces, with the distance function defined by $d(\mathbf{x},\mathbf{y})=|\mathbf{x}-\mathbf{y}|,\mathbf{x},\mathbf{y}\in {\mathbb{R}}^k$.

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Definitions (à la Kulkarni)

Open/Closed ball

Definition 2(Definition).

If $x\in X$ where $(X,d)$ is a metric space and $r\in \mathbb{R},r>0$ , the open (or closed) ball $B$ with center at $x$ and radius $r$ is defined to be the set of all $y\in X$ such that $d(x,y)<r$(or $d(x,y)\le r$). These are analogues of open (or closed) intervals in $\mathbb{R}$.

Neighborhood

Definition 3(Definition).

A neighborhood of $p\in X$ in a metric space $(X,d)$ is any set $S\subseteq X$ which contains an open ball centered at $p$ with radius $r>0$. Basically, elements of $X$ not in the neighborhood should not get arbitrarily close to $p$.

Warning

Rudin’s neighborhood is Kulkarni’s open ball.

Diameter

Definition 4(Definition).

In a metric space $(X,d)$, for $S\subseteq X$, $\text{diam}S\equiv \sup \{d(p,q)|p,q\in S\}$. The supremum is taken in the extended real number system, allowing the diameter to be $\infty$.

Boundedness

Definition 5(Definition).

In a metric space $(X,d)$, $E\subseteq X$ is bounded if there is a real number $M$ and a point $q\in X$ such that $d(p,q)<M$ for all $p\in E$.