# 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!

Related Calculus and Beyond Homework Help News on Phys.org
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