Question: Let $X,Y$ be topological spaces, $x \in X, y \in Y$, and let $\mathscr{N}_x, \mathscr{N}_y$ denote the neighborhood systems of $x$ in $X$ and $y$ in $Y$ respectively. Then is there a criterion to say that $\mathscr{N}_x \cong \mathscr{N}_y$?
Any suggestions or pointers to references which explain or discuss such a criterion (or criteria, or multiple notions of equivalence, etc.) would be sufficient for an answer.
Example: Let $G$ be a topological group, then "intuitively" we should have for all $g \in G$ that $\mathscr{N}_g \cong \mathscr{N}_e$ using translation. In particular, $G$ is first countable if and only if the neighborhood system of the identity $\mathscr{N}_e$ has a countable basis.
Attempt: Analogous to the notion of homeomorphism, it occurs to me to try to use the notion of continuity at a point. One says that a function $f: X \to Y$ is continuous at $x \in X$ iff, for every neighborhood of $f(x) \in Y$, the preimage under $f$ contains a neighborhood of $x$. (See also.)
Now let $f: X \to Y, g: Y \to X$, not necessarily continuous, not necessarily sections/retractions/inverses, such that $f(x)=y$ and $g(y)=x$. Then would it make sense to say that $\mathscr{N}_x \cong \mathscr{N}_y$ if and only if there exist such $f,g$ with $f$ continuous at $x$ and $g$ continuous at $y$?
Such a relation is obviously reflexive and symmetric, transitivity follows from the claim:
If $h: X \to Y$ is continuous at $x \in X$ and $\ell: Y \to Z$ is continuous at $h(x)$, then $\ell \circ h: X \to Z$ is continuous at $x$.
Note: Perhaps the definition should be like that of germs-- I think that such a similarity might follow from the above, but I'm not sure.
