Metric space

A normed space is a special case of a metric space, to see the relationships between metric spaces and others see: Subtypes of topological spaces

Definition of a metric space

A metric space is a set

$$X$$ coupled with a "distance function":

• $$d:X\times X\rightarrow\mathbb{R}$$ or sometimes
• $$d:X\times X\rightarrow\mathbb{R}_+$$ , Note that here I prefer the notation $$d:X\times X\rightarrow\mathbb{R}_{\ge 0}$$

With the properties that for

$$x,y,z\in X$$ :

1. $$d(x,y)\ge 0$$ (This is implicit with the $d:X\times X\rightarrow\mathbb{R}_{\ge 0}$ definition)
2. $$d(x,y)=0\iff x=y$$
3. $$d(x,y)=d(y,x)$$ - Symmetry
4. $$d(x,z)\le d(x,y)+d(y,z)$$ - the Triangle inequality

We will denote a metric space as

$$(X,d)$$ (as $$(X,d:X\times X\rightarrow\mathbb{R}_{\ge 0})$$ is too long and Mathematicians are lazy) or simply $$X$$ if it is obvious which metric we are talking about on $$X$$

Examples of metrics

Euclidian Metric

The Euclidian metric on

$$\mathbb{R}^n$$ is defined as follows: For $$x=(x_1,...,x_n)\in\mathbb{R}^n$$ and $$y=(y_1,...,y_n)\in\mathbb{R}^n$$ we define the Euclidian metric by:

$$d_{\text{Euclidian}}(x,y)=\sqrt{\sum^n_{i=1}((x_i-y_i)^2)}$$

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Proof that this is a metric

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TODO:

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Discrete Metric

Let $X$ be a set. The discrete metric, or trivial metric is the metric defined as follows:

• $$d:X\times X\rightarrow \mathbb{R}_{\ge 0}$$ with $$d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\1 & \text{otherwise}\end{array}\right.$$

However any strictly positive value will do for the $x\ne y$ case. For example we could define $d$ as:

• $$d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\v & \text{otherwise}\end{array}\right.$$
  • Where $v$ is some arbitrary member of $\mathbb{R}_{> 0}$ - traditionally (as mentioned) $v=1$ is used.
    

Note: however in proofs we shall always use the case $v=1$ for simplicity

Notes

Property	Comment
induced topology	discrete topology - which is the topology $(X,\mathcal{P}(X))$ (where $\mathcal{P}$ denotes power set)
Open ball	$B_r(x):=\{p\in X\vert\ d(p,x)< r\}=\left\{\begin{array}{lr}\{x\} & \text{if }r\le 1 \\ X & \text{otherwise}\end{array}\right.$
Open sets	Every subset of $X$ is open. Proof outline: as for a subset $A\subseteq X$ we can show $\forall x\in A\exists r[B_r(x)\subseteq A]$ by choosing say, that is $A$ contains an open ball centred at each point in $A$.
Connected	The topology generated by $(X,d_\text{discrete})$ is not connected if $X$ has more than one point. Proof outline: Let $A$ be any non empty subset of $X$, then define $B:=A^c$ which is also a subset of $X$, thus $B$ is open. Then $A\cap B=\emptyset$ and $A\cup B=X$ thus we have found a separation, a partition of non-empty disjoint open sets, that separate the space. Thus it is not connected if $X$ has only one point then we cannot have a partition of non empty disjoint sets. Thus it cannot be not connected, it is connected.

See also

• Topology induced by a metric
• Connected space
• Topological space