I think the following property is necessary for the converse part of Theorem 4.2. (N-z) $\mathcal{U}_x\neq\emptyset$ for any $x\in X$. (Willard)

Question

I am reading "General Topology" by Stephen Willard.

I think the following property is necessary for the converse part of Theorem 4.2.

N-z) $\mathcal{U}_x\neq\emptyset$ for any $x\in X$.

I created a python program which computes the number of the functions $\mathcal{U}:X=\{0,1,\dots,n-1\}\to 2^X$ which satisfy N-a) through N-d).
But my python program didn't compute the correct answer.

Am I right?

Answer 1

You are correct. The neighborhood system must be nonempty at each point.

If $\mathcal{U}_x$ was empty for some $x\in X$, it means $x$ does not have any neighborhoods, and consequently by the condition N-e) $x$ can not belong to any open set. Especially this means $X$ is not open, but then ''the result is a topology on $X$'' does not hold.

Answer 2

Let $\mathcal{N}_x=\{U\subseteq X| U\in \mathcal{T}_X$ and $x\in U\}$. It’s easy to check $\mathcal{N}_x\neq \emptyset$, since $X\in \mathcal{N}_x$, for all $x\in X$.