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