Advantage of the more general notion of "neighborhoods" in topology

Question

The notion of "open sets" is a fundamental concept in topology. I have been puzzled by (the use of) another slightly more general but closely related one: neighborhoods. Given a topological space $(X,\tau)$, and a point $p\in X$, a neighbourhood of $p$ is a subset $V$ of $X$ that includes an open set $U$ containing $p$, $$ p\in U\subset V. $$

On the other hand, given a "neighborhood system" on a set $X$, one can define a topology that is consistent with the notion of "neighborhoods".

This Wikipedia article makes a remark that "Some mathematicians require that neighbourhoods be open". (For instance, in Munkres's Topology (c.f. page 96), the statement "$U$ is an open set containing $x$" is considered as equivalent to "$U$ is a neighborhood of $x$".)

Question: What is the advantage of the more general notion of "neighborhoods" (that is not required to be open) in practice? Is it simply a matter of taste or does it make significant simplifications in some statement of theorems, proofs or definitions?

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Notes: Please note that this is question is not asking the definitions of "neighborhoods" and "open sets" as the suggested linked question did.

Answer 1

Here is one example when it is convenient not to require neighborhoods to be open. The following is either a lemma or a definition:

A map $f: X\to Y$ of two topological spaces is continuous at a point $x\in X$ if and only if for every neighborhood $V$ of $f(x)$, $f^{-1}(V)$ is a neighborhood of $x$.

Note similarity with the definition of a continuous map.

This lemma/definition will be false if we were to require open neighborhoods. The alternative (when requiring open neighborhoods is heavier) is heavier:

A map $f: X\to Y$ of two topological spaces is continuous at a point $x\in X$ if and only if for every neighborhood $V$ of $f(x)$, $f^{-1}(V)$ contains a neighborhood of $x$.

A historic remark. Bourbaki’s “General Topology” does not require neighborhoods to be open. The convention that neighborhoods are open is common in the US literature and, I think, can be traced to Kelley’s “General Topology.”

Answer 2

It makes for easier formulation of some theorems or definitions: a space can be called locally compact if it has a (base of) compact neighbourhoods, or locally connected if it has a local base of connected neighbourhoods (regardless of openness).

The formulation of local continuity is also easy: $f$ is continuous at $x$ if $f^{-1}[N]$ is a neighbourhood of $x$ for every neighbourhood $N$ of $f(x)$.

$X$ is regular iff every point has a local base of closed neighbourhoods.

A filter $\mathcal{F}$ converges to $x$ iff it contains all neighbourhoods of $x$.