Difference between revisions of "Open set"
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Here <math>(X,d)</math> denotes a metric space, and <math>B_r(x)</math> the [[Open ball|open ball]] centred at <math>x</math> of radius <math>r</math> | Here <math>(X,d)</math> denotes a metric space, and <math>B_r(x)</math> the [[Open ball|open ball]] centred at <math>x</math> of radius <math>r</math> | ||
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==Topology definition== | ==Topology definition== | ||
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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| + | ==References== | ||
| + | {{Definition|Topology|Metric Space}} | ||
Revision as of 14:06, 13 February 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".
Neighborhood
A set N is a neighborhood to a∈X if ∃δ>0:Bδ(a)⊂N
Topology definition
In a topological space the elements of the topology are defined to be open sets
References
- Jump up ↑ Bert Mendelson, Introduction to Topology - definition 6.1, page 52
