(TOPOLOGY) Help understanding definition of open sets

[warped vale]

The definition of open sets is given as:

Let S be a subset of a metric space. Then the set S is open if every point in S has a neighborhood lying in the set.

but can't the same be said of closed sets?

and are those said neighborhood boundary neighborhoods or interior neighborhoods?

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[frail narwhal]

neighbourhoods are open

specifically given $(X,d)$, the open ball of radius $r$ at $x$ is $B_r(x)={y\in X|d(x,y)<r}$.

$S\subseteq X$ is open if for every $x\in S$, there exists $r>0$ such that $B_r(x)\subseteq S$

[rotund solar]

Omegabet_

[thin current]

A neighbourhood of a point x contains an open set around x but is not itself necessarily open.

[warped vale]

thank you guys

+close