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Math and science::Analysis

The Cantor Set

The Cantor set is important for understanding Topology (open and closed sets) and measure.

Cantor set

Let \( C_0 \) be the closed interval \( [0, 1] \) and define \( C_1 \) to be the set that results when the open middle third is removed:

[\[ C_1 = \; C_0 \setminus \; ? = \;\; ?\; \cup \; ?  \]]

This process continued gives us the Cantor set:

\[ C = \bigcap_{n=0}^{\infty} C_n \]
Or alternatively:
[\[ C = [0,1] \setminus \big( (\frac{1}{3},\frac{2}{3}) \; \cup \; (?,\,?) \; \cup \;  (?,\,?) \; \cup \; ...\big) \] ]


Properties of the Cantor set

Below are some properties of the Cantor set, with explanations on the reverse side.

The cantor set...

  • has [zero/finite/infinite] measure
  • is [countably/uncountably] infinite
  • has dimension of [...]
  • is [open/closed/neither/both]
  • is [compact/not compact]
  • is [perfect/not perfect]
  • is [...] dense