Difference between revisions of "Discrete metric and topology"
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==Metric space definition== | ==Metric space definition== | ||
{{:Discrete metric and topology/Metric space definition}} | {{:Discrete metric and topology/Metric space definition}} | ||
| − | ==Open balls== | + | ===Open balls=== |
The [[Open ball|open balls]] of {{M|X}} with the discrete topology are entirely {{M|X}} or a single point, that is: | The [[Open ball|open balls]] of {{M|X}} with the discrete topology are entirely {{M|X}} or a single point, that is: | ||
{{Begin Theorem}} | {{Begin Theorem}} | ||
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This completes the proof | This completes the proof | ||
{{End Proof}}{{End Theorem}} | {{End Proof}}{{End Theorem}} | ||
| − | ==Open sets== | + | ===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))}} | 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}} | {{Begin Theorem}} | ||
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{{Begin Proof}} | {{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|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}} | {{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}} | ||
==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
Revision as of 20:54, 23 July 2015
Contents
[hide]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
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
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
