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 $X$ to 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
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 .
You may be confused about what the statement means. It does not mean that the sets you give as neighborhoods will become the open sets.
Rather, what it says is given a set $X$, that the following two pieces of information are equivalent (in that one of them will uniquely determine the other):
A collection of subsets $\tau$ of $X$ that (i) includes $\varnothing$ and $X$; (ii) is closed under arbitrary unions; and (iii) is closed under finite intersections. (That is, a topology on $X$ given by the collection of "open sets").
For each $x\in X$, a (non-empty) family $\mathcal{N}_x$ of subsets of $X$ such that: (i) For every $A\in \mathcal{N}_x$, $x\in A$; (ii) if $A\in \mathcal{N}_x$ and $A\subseteq B$, then $B\in\mathcal{N}_x$; (iii) if $A,B\in\mathcal{N}_x$, then $A\cap B\in\mathcal{N}_x$; and (iv) For every $A\in \mathcal{N}_x$ there exists $B\in\mathcal{N}_x$ such that for all $y$, if $y\in B$, then $A\in\mathcal{N}_y$. (That is, $\mathcal{N}_x$ is a "system of neighborhoods for each $x\in X$").
So the statement is that given a topology $\tau$ as in 1, there is a way of defining the collection of families as in 2; that given a family of sets $\mathcal{N}_x$ as in 2, there is a way to define a topology $\tau$ using that information; and that if you use $\tau$ to construct the neighborhoods and then use the neighborhoods to construct a topology, you get back the $\tau$ you started with; and if you start with neighborhoods, use them to construct $\tau$, and then you use $\tau$ to construct the neighborhoods, you get back the neighborhoods you started with.
The constructions are as follows: given a topology $\tau$, the family $\mathcal{N}_x$ consists precisely of the set $A$ for which there exists $\mathcal{O}\in\tau$ with $x\in\mathcal{O}$ and $\mathcal{O}\subseteq A$.
And given a collection of families as in 2, the topology $\tau$ is defined to be the collection of all subset $\mathcal{O}$ of $X$ such that for every $x$, if $x\in\mathcal{O}$, then $\mathcal{O}\in\mathcal{N}_x$.
It is a good exercise to verify these constructions have the properties I described in the paragraph after item 2.
