Metric

Definition: Metric

Let $M$ be a set.

A metric on $M$ is a real-valued function which has the following properties for all $x,y,z\in M$:

• Identity of indiscernibles: $d(x,y)\ge 0$ with $d(x,y)=0⟺x=y$

• Symmetry: $d(x,y)=d(y,x)$

• Triangle inequality: $d(x,z)\le d(x,y)+d(y,z)$

INTUITION

A metric on a set can be thought of as a way to define distance between the members of the set.

Definition: Open Ball

Let $(M,d)$ be a metric space.

The open ball of radius $r$ around a given $x\in M$ is the set of all elements in $M$ which are a distance less than $r$ from $m$.

$$\{x\in M\mid d(x,m)<r\}$$

NOTATION

$$B_r(x)B(x,r)$$

Metric Spaces

Definition: Metric Space

A metric space $(M,d)$ is a set $M$ equipped with a metric on it.

The Metric Topology

Theorem: The Metric Topology

Let $(M,d)$ be a metric space.

The collection of all open balls in $M$ forms a base $(M,{\tau }_d)$.

PROOF

TODO

Definition: Metric Topology

The topology ${\tau }_d$ is known as the metric topology induced on $M$ by $d$.

Corollary: Open Sets in the Metric Topology

Let $(S,d)$ be a metric space and let $\tau$ be the topology induced on it by $d$.

A subset $U$ of $S$ is open if and only if for each $u\in U$ there exists an open ball $B_{\epsilon }(u)$ which is contained entirely in $U$.

PROOF

This follows directly from topology generation.

Properties

Theorem: Metric Spaces are Hausdorff Spaces

Every metric space is a Hausdorff Spaces.

PROOF

TODO

Theorem: First-Countability of Metric Spaces

Every metric space is first countable.

PROOF

TODO

Theorem: Convergence of Sequences in Metric Spaces

Let $(M,d)$ be a metric space with its metric topology.

A sequence $(x_n{)}_{n\in I}$ converges to some $l\in M$ if and only if for each open ball $B_{\epsilon }(l)$ around $l$, there exists some $N\in I$ such that $x_n\in B_{\epsilon }(l)$ for all $n\ge N$.

PROOF

TODO