Is countable base basis for topology

Question

I am confused with the definition of countable base,it's definition in is as follows "A space X is said to have a countable base at x if there is a countable collection B of neighborhoods of x such that each neighborhood of x contains at least one element of the elements of B" Such collection may or may not be basis for topology,then In which context we are using the word BASIS here?

Answer 1

The term defines a new concept: that of a local base, a base at a point. Just like a base (not basis; that's an algebraic concept) for a topology can be used to describe all open sets of a space, so can a local base at $x$ be used to understand the structure of open sets that contain a specific $x$.

There is a simple relation two way: if $\mathcal{B}$ is a base for $X$ and $x \in X$ then by definition $\mathcal{B}_x:= \{B \in \mathcal{B}: x \in B\}$ is a local base at the point $x$. So a base contains a lot of local bases. On the other hand if for each $x \in X$ we have a local base $\mathcal{B}_x$ at $x$, then $\mathcal{B}=\bigcup_{x \in X} \mathcal{B}_x$ is a base for the topology of $X$; all local bases together at all points do form a base.

In many spaces we have that at all points we have a countable local base (like in metric spaces where $\{B(x,\frac{1}{n})\mid n \in \mathbb{N}^+\}$ is a local base at $x$), and these are called first-countable (or $C_I$) and more rarely we have that the whole space has a countable base for the topology (like $\mathbb{R}^n$ and its subspaces in their usual topologies) and such spaces are called second countable or $C_{II}$.