Open set

Here

$$(X,d)$$ denotes a metric space, and $$B_r(x)$$ 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" and neighborhood is as you'd expect, "a small area around".

Neighborhood

A set

$$N$$ is a neighborhood to $$a\in X$$ if $$\exists\delta>0:B_\delta(a)\subset N$$

Topology definition

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