Open neighbourhood

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

  1. [ilmath]\exists U\in\mathcal{J}[x\in U\wedge U\subseteq N][/ilmath] for some authors, and us, however for others:
  2. [ilmath]N\in\mathcal{J} [/ilmath] ([ilmath]N[/ilmath] is itself open) and [ilmath]x\in N[/ilmath] - 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, [ilmath]U[/ilmath], containing [ilmath]x[/ilmath]" is the same as "[ilmath]U[/ilmath] is an open neighbourhood to [ilmath]x[/ilmath]"