Metric Spaces

Metric

Definition: Metric

Let $M$ be a set.

A metric on $M$ is a function $d:M\times M\to \mathbb{R}$ 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: $d(x,z)\le d(x,y)+d(y,z)$

Definition: Distance

We call $d(x,y)$ the distance between $x$ and $y$.

Definition: Metric Space

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

The Metric Topology

Definition: Open Ball

Let $(M,d)$ be a metric space and let $p\in M$.

The open ball of radius $r>0$ around $p$ is the set of all elements in $M$ whose distance from $p$ is less than $r$.

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

NOTATION

$$B_r(p)B(p,r)$$

Theorem: The Metric Topology

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

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

PROOF

TODO

Definition: Metric Topology

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

Definition: Equivalent Metrics

Let $M$ be a set.

We say that two metrics are equivalent if they induce the same metric topology on $M$.

Definition: Metrizable Space

A topological space $(X,\tau )$ is metrizable if there exists a metric $d$ on $X$ such that $\tau$ is the metric topology induced by $d$ on $X$.

Theorem: Open Sets in the Metric Topology

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

A subset $U\subseteq M$ is open in $\tau$ if and only if for each $u\in U$ there exists an open ball $B_{\epsilon }(u)$ such that $B_{\epsilon }(u)\subseteq U$.

PROOF

This follows directly from topology generation.

Theorem: Continuity in Metric Spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces with their respective metric topologies.

A function $f:X\to Y$ is continuous at $p\in X$ if and only if for each open ball $B_{\epsilon }(f(p))$ around $f(p)$ there exists some open ball $B_{\delta }(p)$ around $p$ such that if $x$ is inside $B_{\delta }(p)$, then $f(x)$ is inside $B_{\epsilon }(f(p))$.

$$\forall \epsilon >0,\exists \delta >0:x\in B_{\delta }(p)⟹f(x)\in B_{\epsilon }(f(p))$$

PROOF

TODO

Theorem: Metric Spaces are Hausdorff Spaces

Every metric space is a Hausdorff space.

PROOF

TODO

Theorem: First-Countability of Metric Spaces

Every metric space is first-countable.

PROOF

TODO

Boundedness

Definition: Bounded Subset

Let $(M,d)$ be a metric space with the metric topology induced on it by $d$.

A subset $S\subseteq M$ is bounded if there exists some open ball $B$ such that $S\subseteq B$.

Theorem: Boundedness and Distance

Let $(M,d)$ be a metric space with the metric topology induced on it by $d$.

A subset $S\subseteq M$ is bounded if and only if there exists some $r\in {\mathbb{R}}_{\ge 0}$ such that $d(x,y)<r$ for all $x,y\in S$.

PROOF

TODO