What is the difference and relation between fundamental system of neighborhood and basis of topological space ?
Definition.
A base is a collection $B$ of sets such that every set in the topology can be written as a union of sets in $B$. If $(X,T)$ is a topological space and $x∈X$ is an element in $X$,a subset $N ⊂X$ is called a neighborhood of $x$ if there exists some open set $D$ such that $x ∈ D⊂ N$. A collection $N$ of neighborhoods of $x$ is called a fundamental system of neighborhoods of $x$, if for any neighborhood $M$ of $x$, there exists some neighborhood $N$ in $N$ such that $x ∈ N ⊂ M$.
Fundamental system of neighborhood is also called 'basis of point'. Their definition is different, but seems like a bit relevant.. What is the relation between them ?
A topological base (basis) is a family $B$ of $open$ sets such that every open set is the union of (some) members of $B$.
The usual terminology is that a local base (basis) at $x$ is a family $B_x$ of $open$ sets, each containing $x$, such that if $u$ is open and $x\in u$ then some $b\in B_x$ satisfies $x\in b\subseteq u.$
A fundamental system of nbhds of $x$ is also called a nbhd-base at $x$. I admit that the terminology could be better.
I prefer "base". When studying various kinds of topological vector spaces we often consider an algebraic or analytic type of "basis" (e.g. Hamel basis, Hilbert-space basis, Schauder basis) while simultaneously considering a topological "base".