Set of p-adic integers is homeomorphic to Cantor set; how?

[benorin]

Could somebody explain with due brevity why/how the set of p-adic integers is homeomorphic to the Cantor set less one point for any prime p?

This is a quote from Wikipedia:Cantor Set: "The Cantor set is also homeomorphic to the p-adic integers, and, if one point is removed from it, to the p-adic numbers."

Can somebody explain this simply, I don'y really get p-adic #'s.

P.S. Not homework, don't want a proof, just understanding of it.

 

[matt grime]

Write down the bijection from the traditional representation of the cantor set as the reals in [0,1] with no 1's in the base three expansion to the 2-adics (write backwards and put 1s instead of 2s at all points), it is not a deep topological property we're talking about, just a formal one, a little like the integers with the discrete topology are homeomorphic to the positive integers with the discrete topology.

 

[tacman]

Sure, I can try to explain it briefly. First, let's define what p-adic integers and the Cantor set are.

The p-adic integers are numbers that can be expressed as an infinite series of powers of a prime number, p. For example, in base 10, we can write numbers like 123 as 1*10^2 + 2*10^1 + 3*10^0. In p-adic integers, we would write 123 as 1*5^2 + 2*5^1 + 3*5^0, where p=5. This is just one way of writing it, there are many other ways.

The Cantor set is a mathematical set that is constructed by repeatedly removing the middle third of a line segment, and then removing the middle third of the remaining segments, and so on. This results in a set of points that are infinitely close together, but never touch.

Now, to see why the set of p-adic integers is homeomorphic to the Cantor set, we can think of each p-adic integer as a point on a line. Just like the Cantor set, the p-adic integers are constructed by removing some points (in this case, the points that cannot be expressed as an infinite series of powers of p). This results in a set of points that are infinitely close together, but never touch. This is similar to the Cantor set, where the removed points are infinitely close together, but never touch.

To see why removing one point from the Cantor set results in the p-adic numbers, we can think of the removed point as the point that represents the number 0. In p-adic numbers, the number 0 is represented by the infinite series of 0's (e.g. 0*5^2 + 0*5^1 + 0*5^0). So, removing this point from the Cantor set results in a set of points that are infinitely close together, but never touch, just like the p-adic numbers.

In summary, the set of p-adic integers and the Cantor set are both constructed by removing points in a specific way, resulting in sets of points that are infinitely close together, but never touch. This is why they are homeomorphic. I hope this helps to understand the concept better!