Cantor set

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The Cantor ternary set is the set $\mathscr{C}=\displaystyle\bigcap_{k=1}^{\infty} C_k,$ where $C_1=[0,\frac{1}{3}] \cup [\frac{2}{3},1]$ and $C_{k+1}$ is obtained by $C_{k}$ by removing the (open interval) middle-third of each maximal connected interval of $C_k$. Any space homeomorphic to $\mathscr{C}$ is called a Cantor space.

Cantor set.png


Zero-dimensional perfect compact space is Cantor set