Difference between revisions of "Open set"
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"A set <math>U</math> is open if it is a neighborhood to all of its points"<ref>Bert Mendelson, Introduction to Topology - definition 6.1, page 52</ref> and neighborhood is as you'd expect, "a small area around". | "A set <math>U</math> is open if it is a neighborhood to all of its points"<ref>Bert Mendelson, Introduction to Topology - definition 6.1, page 52</ref> 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> | 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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| + | That is if we can puff up any [[Open ball|open ball]] about {{M|x}} that is entirely contained in {{M|N}} | ||
==Topology definition== | ==Topology definition== | ||
Revision as of 23:32, 8 March 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]
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
References
- ↑ Bert Mendelson, Introduction to Topology - definition 6.1, page 52
