Showing the standard topology on $\mathbb{R}$ has a countable basis

Question

I would like to show that the standard topology on $\mathbb{R}$ has a countable basis.

As far as I understand, the standard topology on $\mathbb{R}$ is generated by open intervals. So I can construct one basis as $\{(i,i+1),(i+1/2,i+3/2)\}$ for $i\in\mathbb{Z}$ which is clearly countable.

EDITED: Thanks to Mees de Vries for pointing it out that my example is not a base. So is there any better way to construct a base?

Is it okay to construct one basis to show the existence of a basis? Or is there way to show in general without constructing a specific example?

Could somebody please give some light on this? Thanks.

Answer 1

Your set isn't a basis. If you intersect $(1,2)$ and $(3/2,5/2)$ (using $i=1$) you get $(3/2,2)$ which does not contain any of your "basis" elements.

If you use your set as a sub-basis, it still isn't enough because you can't generate open intervals shorter than 1/2 in length.

The standard approach to finding a countable basis for $\mathbb{R}$ (or $\mathbb{R}^n$) is to use the rational numbers.

Since $\mathbb{Q}$ is countable so is $\mathbb{Q}^2$ (countable cartesian product countable is still countable). Consider open intervals with rational endpoints: $$B = \{ (r,s) \;|\; r,s \in \mathbb{Q} \}$$ Then $B$ has the same cardinality as $\mathbb{Q}^2$ (i.e. it's countable).

This forms a basis for $\mathbb{R}$'s standard topology. To see this notice that these sets are open intervals (thus open in $\mathbb{R}$'s topology). So every open set generated by this basis is a standard open set. Conversely, if $O$ is open in $\mathbb{R}$ and $x \in O$ then there is an open interval $(a,b) \subseteq O$ such that $x \in (a,b)$. Thus $a<x<b$. Pick any rational number between $a$ and $x$ (say $r$) and any rational number between $x$ and $b$ (say $s$). Then $a<r<x<s<b$ and so $x \in (r,s) \subseteq (a,b) \subseteq O$. So $O$ is open relative to our new base $B$. Thus $B$ generates the standard topology.

Answer 2

A basis for a topology on $\mathbb{R}$ is a set $B$ of open sets $B_i$, such that every open set $U$ in $\mathbb{R}$ contains some $B_i$.

To see that the standard topology on $\mathbb{R}$ has a countable base, let $U$ be any open set. Then, let $x \in U$, then $x$ has an open interval around it contained in $U$ i.e. $(x-\delta,x+\delta) \subset U$ for some $\delta > 0$ small. Then, note that there exist rationals $x- \delta < r < s < x+\delta$, hence $(r,s) \subset (x-\delta,x+\delta) \subset U$.

So, every open set contains an open interval with rational endpoints. I want you to prove by yourself that the number of such intervals $(r,s) : r,s \in \mathbb{Q}$ is countable using the fact that $\mathbb{Q}$ is countable. Hence, we have a countable basis for the standard topology on $\mathbb{R}$.

Answer 3

A more conceptual approach: Suppose the metric space $(M,d)$ admits a countable dense set $A$. Then the collection of open balls: $$ \left\{ B\left(a,\frac{1}{n}\right) \ : a\in A, \ n\geq 1 \right\} $$ is countable and gives a basis for the metric topology (of course, both these claims should be proven). In the present context ${\Bbb Q}$ would do for the dense set. But the same proof works e.g. for ${\Bbb R}^d$.