Give \([0, 1]^\omega\) the uniform topology. Find an infinite subset of this space that has no limit point.
Solution
For each \(n\ge1\), let \[ e_n=(0,\dots,0,1,0,\dots)\in[0,1]^\omega \] with the \(1\) in the \(n\)-th coordinate, and set \[ A=\{e_n:n\in\mathbb Z_+\}. \] For \(m\ne n\), the uniform distance is \[ \bar\rho(e_m,e_n)=1, \] since in coordinate \(m\) the two sequences differ by \(1\). Hence the open uniform balls \[ B_{\bar\rho}(e_n,1/3) \] are pairwise disjoint.
More generally, no point \(x\in[0,1]^\omega\) can be a limit point of \(A\). If \(x\) were a limit point, every ball \(B(x,1/3)\) would contain infinitely many \(e_n\). Pick distinct \(e_m,e_n\) in this ball. Then the triangle inequality gives \[ 1=\bar\rho(e_m,e_n)<2/3, \] a contradiction. Thus \(A\) is infinite and has no limit point.