Difference between revisions of "Open set"

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In a [[Topological space|topological space]] the elements of the topology are defined to be open sets
 
In a [[Topological space|topological space]] the elements of the topology are defined to be open sets
  
 
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===Neighbourhood===
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A subset {{M|N}} of a [[Topological space]] {{M|(X,\mathcal{J})}} is a '''neighbourhood of {{M|p}}'''<ref>Introduction to topology - Third Edition - Mendelson</ref> if:
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* <math>\exists U\in\mathcal{J}:p\in U\wedge U\subset N</math>
 
==See also==
 
==See also==
 
* [[Closed set]]
 
* [[Closed set]]
  
 
==References==
 
==References==
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<references/>
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{{Definition|Topology|Metric Space}}
 
{{Definition|Topology|Metric Space}}

Revision as of 18:48, 19 April 2015


Here (X,d) denotes a metric space, and Br(x) the open ball centred at x of radius r

Metric Space definition

"A set U is open if it is a neighborhood to all of its points"[1] and neighborhood is as you'd expect, "a small area around".

Neighbourhood

A set N is a neighborhood to aX if δ>0:Bδ(a)N

That is if we can puff up any open ball about x that is entirely contained in N

Topology definition

In a topological space the elements of the topology are defined to be open sets

Neighbourhood

A subset N of a Topological space (X,J) is a neighbourhood of p[2] if:

  • UJ:pUUN

See also

References

  1. Jump up Bert Mendelson, Introduction to Topology - definition 6.1, page 52
  2. Jump up Introduction to topology - Third Edition - Mendelson