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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?

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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 $C_{II}$.

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    $\begingroup$ 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. $\endgroup$ – DanielWainfleet Feb 7 at 23:21
  • $\begingroup$ @DanielWainfleet Engelking does not, and that's the book I mostly use. $\endgroup$ – Henno Brandsma Feb 7 at 23:22
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    $\begingroup$ I have Engelking. What a masterpiece. My comment was meant more for the proposer. $\endgroup$ – DanielWainfleet Feb 8 at 19:23

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