Consider the set $D=\{\frac{1}{n}:n=1,2,…\}\cup\{0\}$ and let $X=C(D,\mathbb{R})$ with $d_X(f,g)=\sup_{x\in A}|f(x)-g(x)|$. Show that $X$ and $l^\infty(\mathbb{N})$ are not homeomorphic.
I tried to find a topological property not shared by two spaces but I cannot find one. They are both metric spaces. Is there any other methods I can try?
Best Answer
The space $C(D,\mathbb R)$ can be identified with the space $c$ of all convergent sequences of real numbers with the sup norm. Sequences of rational numbers which are eventually constant form a countable dense set. Since $l^{\infty}$ is not separable, the two spaces are not homeomorphic.