Difference between revisions of "Discrete metric and topology"

Latest revision as of 18:10, 25 April 2017

Metric space definition

Let $X$ be a set. The discrete^[1] metric, or trivial metric^[2] 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)↦{0if x=yvotherwise
- Where $v$ is some arbitrary member of $\mathbb{R}_{> 0}$ ^{[Note 1]} - traditionally (as mentioned) $v=1$ is used.

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

Metric summary

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.

Metric objects

Open balls

The open balls of $X$ with the discrete topology are entirely $X$ or a single point, that is:

[Expand]
$B_r(x):=\{p\in X\vert\ d(x,p)<r\}=\left\{\begin{array}{lr}\{x\} & \text{for }r\le 1\\ X & \text{otherwise}\end{array}\right.$

By definition

$B_r(x):=\{p\in X\vert\ d(x,p)<r\}$ note that for:

r≤1 we have Br(x)={x} as
- $d(x,p)< r \le 1$ so $d(x,p)<1$ only when $x=p$ , as if $x\ne p$ then $d(x,p)=1\not<1$ (proof by contrapositive)
r> 1 we have B_r(X)=X as
- $d(x,y)\le 1$ always, so we have $\forall x,y\in X[d(x,y)\le 1< r]$ so $\forall x,y\in X[d(x,y)< r]$ thus the ball contains every point in $X$

This completes the proof

Open sets

The open sets of $(X,d_\text{discrete})$ consist of every subset of $X$ (the power set of $X$ ) - this is how the topology induced by the metric may be denoted $(X,\mathcal{P}(X))$

[Expand]
Every subset of $X$ is an open set

Let

$A$ be a subset of

$X$ , we will show that

$\forall x\in A\exists r>0[B_r(x)\subseteq A]$

Let x\in A be given
- Choose r=\tfrac{1}{2}
  Now B_r(x)=\{x\}\subseteq A
  - This must be true as we know already that $x\in A$ (to show this formally use the implies-subset relation)
- We have shown that given an $x\in A$ we can find an open ball about $x$ entirely contained within $A$
We have shown that for any $x\in A$ we can find an open ball about $x$ entirely contained within $A$

This completes the proof

Discrete topology

The discrete topology on $X$ is the topology that considers every subset to be open. We may write $X$ imbued with the discrete topology as:

$(X,\mathcal{P}(X))$ where $\mathcal{P}$ denotes power set

TODO: find reference - even though it is obvious as I show above that every subset is open

Related theorems

Notes

Jump up ↑ Note the strictly greater than 0 requirement for $v$

References

Jump up ↑ Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene
Jump up ↑ Functional Analysis - George Bachman and Lawrence Narici

[3] Jump up ↑ Note the strictly greater than 0 requirement for $v$

[ITTGG-1] Jump up ↑ Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene

[2] Jump up ↑ Functional Analysis - George Bachman and Lawrence Narici

[1]

[2]

[Note 1]

@@ Line 1: / Line 1: @@
+__TOC__
 ==Metric space definition==
 {{:Discrete metric and topology/Metric space definition}}
+==Metric summary==
+{{:Discrete metric and topology/Summary}}
+==Metric objects==
+===Open balls===
+The [[Open ball|open balls]] of {{M|X}} with the discrete topology are entirely {{M|X}} or a single point, that is:
+{{Begin Theorem}}
+* {{MM|1=B_r(x):=\{p\in X\vert\ d(x,p)<r\}=\left\{ $\begin{array}{lr}\{x\} & \text{for }r\le 1\\ X & \text{otherwise}\end{array}$ \right.}}
+{{Begin Proof}}
+: By definition {{MM|1=B_r(x):=\{p\in X\vert\ d(x,p)<r\} }} note that for:
+:* {{M|1=r\le 1}} we have {{M|1=B_r(x)=\{x\} }} as
+:**  {{M|1=d(x,p)< r \le 1}} so {{M|1=d(x,p)<1}} only when {{M|1=x=p}}, as if {{M|x\ne p}} then {{M|1=d(x,p)=1\not<1}} (proof by [[Contrapositive|contrapositive]])
+:* {{M|1=r> 1}} we have {{M|1=B_r(X)=X}} as
+:** {{M|d(x,y)\le 1}} always, so we have {{M|\forall x,y\in X[d(x,y)\le 1< r]}} so {{M|\forall x,y\in X[d(x,y)< r]}} thus the ball contains every point in {{M|X}}
+This completes the proof
+{{End Proof}}{{End Theorem}}
+===Open sets===
+The [[Open set|open sets]] of {{M|(X,d_\text{discrete})}} consist of every subset of {{M|X}} (the [[Power set|power set]] of {{M|X}}) - this is how the [[Topology induced by a metric|topology induced by the metric]] may be denoted {{M|(X,\mathcal{P}(X))}}
+{{Begin Theorem}}
+Every subset of {{M|X}} is an open set
+{{Begin Proof}}
+: Let {{M|A}} be a subset of {{M|X}}, we will show that {{M|\forall x\in A\exists r>0[B_r(x)\subseteq A]}}
+:* Let {{M|x\in A}} be given
+:** Choose {{M|1=r=\tfrac{1}{2} }}
+:**: Now {{M|1=B_r(x)=\{x\}\subseteq A}}
+:**:* This must be true as we know already that {{M|x\in A}} (to show this formally use the [[Implies and subset relation|implies-subset relation]])
+:** We have shown that given an {{M|x\in A}} we can find an open ball about {{M|x}} entirely contained within {{M|A}}
+:* We have shown that for any {{M|x\in A}} we can find an open ball about {{M|x}} entirely contained within {{M|A}}
+This completes the proof
+{{End Proof}}{{End Theorem}}
+==Discrete topology==
+The ''discrete topology'' on {{M|X}} is the [[Topological space|topology]] that considers every subset to be open. We may write {{M|X}} imbued with the discrete topology as:
+* {{M|(X,\mathcal{P}(X))}} where {{M|\mathcal{P} }} denotes [[Power set|power set]]
+{{Todo|find reference - even though it is obvious as I show above that every subset is open}}
+==Related theorems==
+* [[Every map from a space with the discrete topology is continuous]]
+* [[Every continuous map from a non-empty connected space to a discrete space is constant]]
+==Notes==
+<references group="Note"/>
+==References==
+<references/>
 {{Definition|Topology|Metric Space}}

Difference between revisions of "Discrete metric and topology"

Latest revision as of 18:10, 25 April 2017

Contents

Metric space definition

Metric summary

Metric objects

Open balls

Open sets

Discrete topology

Related theorems

Notes

References

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