Can we live without neighborhood basis but with open neighborhood basis?

Question

I am reading Lee's Introduction to Topological Manifolds, and he declares that neighborhoods always mean open neighborhoods.

So, the definition of a open neighborhood basis goes:

Def Let $X$ be a topological space and $p \in X$. The collection $\mathcal{B}_p$ of neighborhoods of p is called neighborhood basis for $X$ at $p$ if every neighborhood contains some $B \in \mathcal{B}_p$.

This only cares about open neighborhood basis. Is this definition has any defect, I mean is there any mathematical concept that cannot be expressed as an open neighborhood basis, and needs neighborhood basis? If not, why not every mathematician follow this convention?

Answer 1

Let $(X,\tau)$ be a topological space, and $p \in X$.

• an "open neighbourhood" of $p$ is a set $V \in \tau$ such that $p \in V$.
• a "neighbourhood" of $p$ is a set $V \subset X$ such that exists $\Omega \in \tau$, with $x \in \Omega \subset V$.

There's no harm, because:

• every "open neighbourhood" is a "neighbourhood" itself;
• every "neighbourhood" contains an "open neighbourhood".

In many results, it is easier to work assuming that the neighbourhood is already open. So some people prefer to include openness in the definition, to avoid the hassle.