# The empty set is a neighborhood?

The following axioms of a Topological space is from Wikipedia:

Neighbourhoods definition

This axiomatization is due to Felix Hausdorff. Let $$X$$ be a set; the elements of $$X$$ are usually called points, though they can be any mathematical object. We allow $$X$$ to be empty. Let $$N$$ be a function assigning to each $$x$$ (point) in $$X$$ a non-empty collection $$N(x)$$ of subsets of $$X$$. The elements of $$N(x)$$ will be called neighbourhoods of $$x$$ with respect to $$N$$ (or, simply, neighbourhoods of $$x$$). The function $$N$$ is called a neighbourhood topology if the axioms below are satisfied; and then $$X$$ with $$N$$ is called a topological space.

1. If $$N$$ is a neighbourhood of x (i.e., $$N \in N(x)$$), then $$x \in N$$. In other words, each point belongs to every one of its neighbourhoods.
2. If $$N$$ is a subset of $$X$$ and contains a neighbourhood of $$x$$, then $$N$$ is a neighbourhood of $$x$$. I.e., every superset of a neighbourhood of a point $$x$$ in $$X$$ is again a neighbourhood of $$x$$.
3. The intersection of two neighbourhoods of $$x$$ is a neighbourhood of $$x$$.
4. Any neighbourhood $$N$$ of $$x$$ contains a neighbourhood $$M$$ of $$x$$ such that $$N$$ is a neighbourhood of each point of $$M$$.

Then there is an example given:

Examples

1. $$X = \{1, 2, 3, 4\}$$ and collection $$\tau = \{\{\}, \{1, 2, 3, 4\}\}$$ of only the two subsets of $$X$$ required by the axioms form a topology, the trivial topology (indiscrete topology).
2. $$X = \{1, 2, 3, 4\}$$ and collection $$\tau = \{\{\}, \{2\}, \{1, 2\}, \{2, 3\}, \{1, 2, 3\}, \{1, 2, 3, 4\}\}$$ of six subsets of $$X$$ form another topology.
3. $$X = \{1, 2, 3, 4\}$$ and collection $$\tau = P(X)$$ (the power set of $$X$$) form a third topology, the discrete topology.
4. $$X = \mathbb{Z}$$, the set of integers, and collection $$\tau$$ equal to all finite subsets of the integers plus $$\mathbb{Z}$$ itself is not a topology, because (for example) the union of all finite sets not containing zero is infinite but is not all of $$\mathbb{Z}$$, and so is not in $$\tau$$.

Here is my question: From axiom 1, it is said that an element $$x$$ in the set must belong to any of its neighborhoods, but in examples 1 and 2, the empty set is a neighboorhood. I don't understand it. I am not a math student, I'm just curious.

• Those examples are from the "open sets definition" section of the page. The sets in those examples are the open sets of a topology, not its neighborhoods. Commented Aug 20, 2015 at 4:16

The empty set $\varnothing$ is not a neighborhood of any point $x\in X$, because as you correctly observed, there are no elements of $\varnothing$. However, the examples you are citing are not claiming that $\varnothing$ is a neighborhood, so there is no contradiction.
The examples are sets $X$ together with a topology $\tau$, so to say that a subset $U\subseteq X$ satisfies $U\in\tau$ means that $U$ is an open set, not (necessarily) a neighborhood of any particular point $x$.
Given such a structure, we can define a subset $U$ of $X$ to be open if $U$ is a neighbourhood of all points in $U$.
The subset $U=\varnothing$ vacuously satisfies this property, because there are no points in $\varnothing$.