Difference between revisions of "Open set"
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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> | 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> | ||
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| + | ==Topology definition== | ||
| + | In a [[Topological space|topological space]] the elements of the topology are defined to be open sets | ||
Revision as of 14:04, 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- Jump up ↑ Bert Mendelson, Introduction to Topology - definition 6.1, page 52
