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Cantor Set Homeomorphic to Inverse Limit Space

joypav

Active member
Mar 21, 2017
151
Problem:
For each positive integer $n$ let $X_n = \left\{1, 2, 3, ..., 2^n \right\}$ with the discrete topology and let $f_n : X_{n+1} → X_n$ be the function defined by:
$f_n(i) = i$ for $1 ≤ i ≤ 2^n$
$f_n(i) = 2^{n+1} − i + 1$ for $2^n < i ≤ 2^{n+1}$
Then
$X = lim_{←} \left\{X_i, f_i \right\}_{i=1}^{\infty}$
is homeomorphic to the Cantor set.


I've been trying to find a homeomorphism for this for way too long... anyone know what it would be?
I can take care of proving it's a homeomorphism once I have the function.