# Neighborhood basis in the discrete metric space and first countability

Consider the topological space $$\mathbb{R}$$ with discrete metric

$$d(x,y) = \begin{cases} 1 & x\neq y\\ 0 & x=y \end{cases}$$

We know that the metric space is first countable. So for each $$x\in \mathbb{R}$$, there exists a countable neighborhood basis at $$x$$.

My question is what is that basis?

I think $$\{\{x\}\}$$ is a local base at each $$x$$, so there is only one element in this basis? How about if I consider the open ball with $$r>1$$, then the number of elements in this basis is infinite. It becomes uncountable.

I am confused about how to find the neighborhood basis of $$x$$.

## 1 Answer

You are right; $$\bigl\{\{x\}\bigr\}$$ is a a countable (finite, actually) system of neighborhoods of $$x$$, for each $$x\in X$$.

Note that, with respect to the discrete topology, $$V$$ is a neighborhood of $$x$$ if and only if $$x\in V$$.

• Could you please say more about neighborhood basis for this discrete topology? thanks! Commented Oct 8, 2018 at 7:21
• That's what my second sentence is about: $\bigl\{\{x\}\bigr\}$ is a neighborhood basis for $x$ since each of its elements is a neighborhood of $x$ and since, for each neighborhood $V$ of $x$, $V\supset\{x\}$. Commented Oct 8, 2018 at 7:28