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

Solution 1:

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 $x$.

Solution 2:

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.”