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 | ||
| − | + | ===Neighbourhood=== | |
| + | 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: | ||
| + | * <math>\exists U\in\mathcal{J}:p\in U\wedge U\subset N</math> | ||
==See also== | ==See also== | ||
* [[Closed set]] | * [[Closed set]] | ||
==References== | ==References== | ||
| + | <references/> | ||
| + | |||
{{Definition|Topology|Metric Space}} | {{Definition|Topology|Metric Space}} | ||
Revision as of 18:48, 19 April 2015
Here [math](X,d)[/math] denotes a metric space, and [math]B_r(x)[/math] the open ball centred at [math]x[/math] of radius [math]r[/math]
Contents
Metric Space definition
"A set [math]U[/math] 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 [math]N[/math] is a neighborhood to [math]a\in X[/math] if [math]\exists\delta>0:B_\delta(a)\subset N[/math]
That is if we can puff up any open ball about [ilmath]x[/ilmath] that is entirely contained in [ilmath]N[/ilmath]
Topology definition
In a topological space the elements of the topology are defined to be open sets
Neighbourhood
A subset [ilmath]N[/ilmath] of a Topological space [ilmath](X,\mathcal{J})[/ilmath] is a neighbourhood of [ilmath]p[/ilmath][2] if:
- [math]\exists U\in\mathcal{J}:p\in U\wedge U\subset N[/math]
