Clarification of topological space definition in terms of neighborhoods

Question

Let $X$ be a set. We allow $X$ to be empty. Let $N$ be a function assigning to each $x$ 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 $x$).

An empty set has no subsets. If $X$ is empty (which the definition says is permissible), and $N$ must be a function assigning to each element $x$ in $X$ a non-empty collection of subsets of $X$, then...what's going on here?

An empty set can have a non-empty collection of subsets? Please elaborate. I'm trying to understand the concept of continuity and uniform continuity in a topological sense and misunderstanding of this definition is kind of in the way.

Thanks.

Answer 1

Here is another way of looking at what others have said. A function $f:X \to Y$ should consist of a triple $(X,Y,G_f)$ where $G_f$ is a "functional" subset of $X \times Y$, in the sense that for each $x \in X$ there is a unique $y$ in $Y$ such that $(x,y) \in G_f$. So if $X$ is empty, there is a unique function $X \to Y$. In categorical terms, the empty set is initial in the category of sets and function, just as the trivial group is initial in the category of groups and morphisms.

Answer 2

This is vacuous when $X$ is empty. It is easy to assign a non-empty subset $N(x)\subseteq X$ to each element $x\in X$ when there are no elements in the set $X$ to begin with!

After all... can you point out an element $x$ which violates the property?

Answer 3

If $X$ is empty, all predicates of the form $\forall x\in X,\ P(x)$ are true. In particular, yes, for all $x\in\emptyset$, the property of $N(x)$ is verified.

Answer 4

You are confused on two counts, and the other answers address only one of them.

First, as the others have said, even in a world with no fire-breathing dragons, it's easy to assign a fire-breathing dragon to each element of the empty set, because there are no elements of the empty set.

But second, as the others have not said, in this case there are fire-breathing dragons. That is, there is a non-empty collection of subsets of the empty set. Namely, the empty set has one subset, namely $\emptyset$. There are, therefore, two collections of subsets of the empty set. There's the empty collection, with no elements at all, and the complete collection that includes $\emptyset$, namely $\lbrace \emptyset\rbrace$. Of these, one is nonempty (namely the second one). So there you have a nonempty collection of subsets of the empty set.