# Is countable base basis for topology

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?

## 1 Answer

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 $$C_{II}$$.

• Some popular texts do use "basis" for "base" This can be confusing, especially regarding topological vector spaces. And it is unfortunate that "bases" is the plural of both words. – DanielWainfleet Feb 7 at 23:21
• @DanielWainfleet Engelking does not, and that's the book I mostly use. – Henno Brandsma Feb 7 at 23:22
• I have Engelking. What a masterpiece. My comment was meant more for the proposer. – DanielWainfleet Feb 8 at 19:23