Alternative way of defining a topology

definitiongeneral-topology

tl;dr: In trying to come up with a way to characterize topologies in terms of neighborhoods (rather than open sets) I run into a weird, not-quite-equivalent-to-topology structure. I wonder if I can fix my definition to give actual topologies, or else what can be said about the structure I have defined.

Definition: Given a set $X$, an n-topology $\tau$ on a set $X$ assigns to each $x \in X$ a set $\tau_x \subseteq \mathcal P(X)$ satisfying the following constraints:

  • $X \in \tau_x$
  • If $N \in \tau_x$ then $x \in N$
  • If $N \in \tau_x$ and $N' \supseteq N$ then $N' \in \tau_x$
  • If $N, N' \in \tau_x$, then $N \cap N' \in \tau_x$.

We call the sets in $\tau_x$ the neighborhoods of $x$.

Clearly, given a topology on a set $X$ we get an n-topology: we take the sets $\tau_x$ to be the neighborhoods of $x \in X$ in the topological sense, as all sets which contain an open set which contains $x$. Conversely, given an n-topology, we obtain a topology by declaring a set open if it is a neighborhood of all of its elements. Furthermore, we can define continuous functions on n-topological spaces by saying that $f : (X, \tau) \to (Y, \sigma)$ is n-continuous if
$$
N \in \sigma_{f(x)} \implies f^{-1}(N) \in \tau_x.
$$
It appears that this notion coincides with our usual notion of continuous.

Going from topology to n-topology to topology, we recover the original topology. However, going from n-topology to topology to n-topology, we might lose information. For example, we can define an n-topology on $\omega$ by stating that $\tau_0$ contains all cofinite sets containing 0, while $\tau_n$ for $n > 0$ contains only $\omega$ itself. This clearly satisfies the above conditions. If we turn it into a topology, we just get the indiscrete topology $\{\emptyset, \omega\}$ on $\omega$, which we could also have gotten by taking $\tau_n = \{\omega\}$ for each $n \in \omega$.

Question: Can I add a condition to this notion of "n-topology" to make it an equivalent definition to ordinary topologies? Is the thing I call an n-topology something interesting — has anyone else used it, and if so under what name?

Best Answer

The idea of axiomatizing the notion of "neighborhood" is not new. Wikipedia has a discussion of an axiomatization which they credit to Felix Hausdorff. He has the same axioms as you plus one more:

  • If $N \in \tau_x$, there exists $M \in \tau_x$ such that $M \subseteq N$ and for all $y \in M$ we have $N \in \tau_y$.

With this additional axiom, you do in fact get an equivalent notion to the usual notion of topology. In particular, using your definition of "open", you have the theorem that every neighborhood of $x$ contains an open neighborhood of $x$. Note that your counterexample on $\omega$ does not satisfy this additional axiom.

Related Question