# Let X:=R^n with the Euclidean Topology. Is X first countable? Find a nested basis

## Homework Statement

Let X:=ℝn with the Euclidean Topology. Is X first countable? Find a nested neighborhood basis for X at 5.

## Homework Equations

If X is a topological space and p$\in$X, a collection $B$p of neighborhoods of p is called a neighborhood basis for X at p if every neighborhood of p contains some B$\in$$B$p.

We say X is first countable if there exists a countable neighborhood basis at each point.

## The Attempt at a Solution

I say yes.
Let p$\in$X, the set of open balls Br(p) for r being rational forms a neighborhood basis at p. (That is, for all neighborhoods U of p, there is a Br(p)$\subseteq$U)
Since p was arbitrary and this $B$p is countable (since rationals are countable), X is first countable.

As well, we can just let the nest interval be defined as: B(1/2)i(5) for i being a natural number. Thus, B(1/2)i+1(5)<B(1/2)i(5).

I am struggling a bit at this level of proof honestly, and I'm trying to stay afloat. Thank you!

jbunniii
Homework Helper
Gold Member

Your proof looks fine. Just a few nitpicking details: first, $r$ needs to be rational AND positive. Second, although it's pretty obvious, you might say a few words about why you can find an $r$ such that $B_r(p) \subseteq U$.

Your nested neighborhood basis at 5 is fine. Just one minor detail: instead of <, you want $\subset$.

Thank you!

Wouldn't it just be based on the density of the rationals?
Also, thank you for catching my typo :)

dextercioby