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".

Neighbourhood

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 $N$

Topology definition

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

Neighbourhood

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

• $$\exists U\in\mathcal{J}:p\in U\wedge U\subset N$$

See also

• Relatively open set
• Closed set
• Relatively closed set