EDB — 0MP

[0MP]Difficulty:*.We restrict the topology described in the previous example to the set \(Y=[0,1]^{[0,1]}\) (that is, we restrict \(ℝ\) to \([0,1]\), and set \(Ω=[0,1]\)). Find a sequence \((f_ n)⊂ Y\) that does not allow a convergent subsequence.

Let’s recall the definition [0J3]↺↻: a space \(X\) is ”compact by coverings” if, for every \((A_ i)_{i∈ I}\) family of open such that \(⋃_{i∈ I}A_ i= X\), there is a finite subfamily \(J⊂ I\) such that \(⋃_{i∈ J}A_ i= X\). The Tychonoff theorem shows that this space \(Y\) is ”compact by coverings”. This exercise shows you instead that \(Y\) it is not ”compact by sequences”.