Difference between revisions of "Discrete metric and topology"
From Maths
(Created page with "==Metric space definition== {{:Discrete metric and topology/Metric space definition}} {{Definition|Topology|Metric Space}}") |
m |
||
| Line 1: | Line 1: | ||
==Metric space definition== | ==Metric space definition== | ||
{{:Discrete metric and topology/Metric space definition}} | {{:Discrete metric and topology/Metric space definition}} | ||
| + | ==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\{{x}for r≤1Xotherwise\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]}} | ||
| + | {{Todo|Do this, it's easy enough see [[Metric space]] page for outline}} | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | |||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Topology|Metric Space}} | {{Definition|Topology|Metric Space}} | ||
Revision as of 20:40, 23 July 2015
Metric space definition
Let X be a set. The discrete[1] metric, or trivial metric[2] is the metric defined as follows:
- d:X×X→R≥0 with d:(x,y)↦{0if x=y1otherwise
However any strictly positive value will do for the x≠y case. For example we could define d as:
- d:(x,y)↦{0if x=yvotherwise
- Where v is some arbitrary member of R>0[Note 1] - traditionally (as mentioned) v=1 is used.
- Where v is some arbitrary member of 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
Open balls
The open balls of X with the discrete topology are entirely X or a single point, that is:
[Expand]
- Br(x):={p∈X| d(x,p)<r}={{x}for r≤1Xotherwise
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
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
