Q:

Topology: definition via neighborhood vs open sets?

I'm reading the Wikipedia page on Topology.  It claims topology can be defined using axioms of neighbors N(x) or open sets.  I'm confused because the neighbor axioms do not allow a topology with the empty set, because x must be included in the topology.  Yet, the definition of open set topology requires the empty set and X belong to the topology.
Can anyone clear up my confusion between neighborhood and open set definitions?  Why do they result in different topologies?
Thank you

A:

The definitions are equivalent, and the empty set is open if you use the definition with neighbourhoods.
Namely, for every element $x$ of the empty set (note: there aren't any!) you can find the whole neigbourhood from $N(x)$ contained in the same empty set (provided that you can show me that element $x$ in the first place!).
In other words, this is true precisely because the premise ($x\in\emptyset$) is always false. See also: https://en.wikipedia.org/wiki/Vacuous_truth .

