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Map from binary sequences on $\{0,1\}$ into the Cantor set $C$ respects order
It is a standard problem in elementary topology to show that the cantor set is homeomorphic to the countable product $\displaystyle\prod \{0,2\}$ (with the product topology of the discrete topology on $\{0,2\}$ ). This is shown via the map $(a_n) \mapsto \displaystyle\sum \frac{a_n}{3^n}$, which is readily seen to be continuous and surjective. My question is
'How does one justify that this map is injective?'
In general, base 3 expansions are not unique. For example,
$1 =1\cdot3^0 + 0\cdot3^1 + 0\cdot3^2 + 0\cdot3^3 + \cdots = 0\cdot3^0 + 2\cdot3^1 + 2\cdot3^2 + 2\cdot3^3 + \cdots$
Is there some basic fact of general base expansion (non)uniqueness that I am overlooking? How does removing the digit '1' from the mix change anything?