- #1

Hodgey8806

- 145

- 3

## Homework Statement

Let X:=ℝ

^{n}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[itex]\in[/itex]X, a collection [itex]B[/itex]

_{p}of neighborhoods of p is called a neighborhood basis for X at p if every neighborhood of p contains some B[itex]\in[/itex][itex]B[/itex]

_{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[itex]\in[/itex]X, the set of open balls B

_{r}(p) for r being rational forms a neighborhood basis at p. (That is, for all neighborhoods U of p, there is a B

_{r}(p)[itex]\subseteq[/itex]U)

Since p was arbitrary and this [itex]B[/itex]

_{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!