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"^[1] and neighborhood is as you'd expect, "a small area around".

A set

$N$ is a neighborhood to

$a\in X$ if

$\exists\delta>0:B_\delta(a)\subset N$

That is if we can puff up any open ball about $x$ that is entirely contained in $N$

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

A subset $N$ of a Topological space $(X,\mathcal{J})$ is a neighbourhood of $p$ ^[2] if:

<cite_references_link_accessibility_label> ↑ Bert Mendelson, Introduction to Topology - definition 6.1, page 52
<cite_references_link_accessibility_label> ↑ Introduction to topology - Third Edition - Mendelson