1. ## The Cantor Set

The hint given for the problem is that the set of isolated pts of a countable complete metric space X forms a dense subset of X, and using this i need to prove that the cantor set is uncountable. so i was thinking that we need to use the fact that since cantor set has no isolated points (which was proven in previous exercise), but i keep getting stuck. can someone help me?

2. Originally Posted by squarerootof2
The hint given for the problem is that the set of isolated pts of a countable complete metric space X forms a dense subset of X, and using this i need to prove that the cantor set is uncountable. so i was thinking that we need to use the fact that since cantor set has no isolated points (which was proven in previous exercise), but i keep getting stuck. can someone help me?
Suppose that $\displaystyle C$, the Cantor set, is countable. Since $\displaystyle (\mathbb{R}, | ~ |)$ is a complete metric space it would mean the isolated points of $\displaystyle C$, call this subset $\displaystyle S$, would be dense in $\displaystyle C$. Topologically it means $\displaystyle \bar S = C$. But $\displaystyle S = \emptyset$ by your previous exercise, so the closure of $\displaystyle S$ is the empty set, but that is a contradiction because it has to be $\displaystyle C$. This means the Cantor set is uncountable.

3. thanks so much for the input, but one small question, when you say "Since is a complete metric space " what is the metric you are using? just the ordinary euclidean metric? thanks.

4. Originally Posted by squarerootof2
thanks so much for the input, but one small question, when you say "Since is a complete metric space " what is the metric you are using? just the ordinary euclidean metric? thanks.
Yes. Here it is the same thing as taking the absolute value of the number.