Open set
From Maths
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
