Open set

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" and neighborhood is as you'd expect, "a small area around".

Neighborhood

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]

Topology definition

In a topological space the elements of the topology are defined to be open sets