Open neighbourhood

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Definition

Recall that if $N$ is a neighbourhood to a point $x\in X$ for a topological space $(X,\mathcal{ J })$ that this means:

1. $\exists U\in\mathcal{J}[x\in U\wedge U\subseteq N]$ for some authors, and us, however for others:
2. $N\in\mathcal{J}$ ($N$ is itself open) and $x\in N$ - we call these "open neighbourhoods", they're neighbourhoods to a point, but must be an open set itself rather than just contain one.

"There exists an open set, $U$, containing $x$" is the same as "$U$ is an open neighbourhood to $x$"