EDB — 0RP

Exercises

1. [0RP] Consider the example of the set \(E⊆ℝ^ 2\) given by

   \begin{equation} E=\big\{ (0,t):\ -1≤ t≤1\big\} ∪\Big\{ \Big(x,\sin \frac 1 x\Big):\ x∈ (0,1]\Big\} \quad .\label{eq:connesso_ ma_ non_ archi} \end{equation}

   65

   Show that this set is closed, connected, but is not path connected.

   Solution 1

   [0RQ]↺↻

   This set is sometimes called closed topologist’s sine curve [ 37 ] .