1st December 2004

Munkres §20

Ex. 20.5. Consider R^{ω} with the uniform topology and let d be the uniform metric. Let C ⊂ R^{ω} be the set of sequences that converge to 0. Then

\overline{R^{∞}} = C.

⊂: Since clearly R^{∞} ⊂ C it is enough to show that C is closed. Let (x_{n}) ∈ R^{ω} − C be a sequence that does not converge to 0. This means that there is some 1 > ε > 0 such that |x_{n}| > ε for infinitely many n. Then B_{d}((x_{n}), \frac{1}{2}ε) ⊂ R^{ω} − C.

⊃: Let (x_{n}) ∈ C. For any 1 > ε > 0 we have |x_{n}| < ε for all but finitely many n. Thus B_{d}((x_{n}), 2ε) ∩ R^{∞} ≠ ∅.

References