I’m teaching topology this semester. The students are looking at different topologies on the real number line. For homework I asked them to think about which topologies are “the same” (if any) and which are “different,” and why they thought that was the case. We haven’t yet talked about continuous maps or homeomorphisms, so I told them that two topologies were the same if we could rename all of the points so that one topology turned into the other.
They came back with the answers I expected. For example, the trivial topology has only two open sets and all the others have infinitely many, so the trivial topology must be different from the rest. One-point sets are open in the discrete topology, but not in any of the others, so it must be different from the rest.
One student pointed out that the discrete topology is the power set of the real numbers, , and we know (by Cantor’s Theorem) that has a different cardinality than . So he wanted to know what the cardinalities of the other topologies were. For example, if the usual Euclidean topology on , which we’ll denote , has the cardinality of the continuum (the cardinality of ), then this would be another way to show that these two topologies aren’t the same.
My first reaction was joy—I was thrilled with this excellent observation. Then I realized that I didn’t know the cardinality of . I told the class that I strongly suspected that it had the cardinality of the continuum, but that I couldn’t say for sure on the spot.
This is probably “a fact well known to those who know it well” (this was a quote from a book I once read), but I wasn’t one of those people. So I decided to prove it (with the help of my colleague Jeff).
Theorem. and have the same cardinality.
By definition, every set in can be written as the union of -balls, . In fact, it is not difficult to see that every open set can be written as the union of -balls with rational center and rational .
In particular, there is a surjective function . Namely, for ,
We know that is countable, and the power set of a countable set has the cardinality of the continuum. Thus there is a bijective function . Hence is surjective.
On the other hand, there is a surjective function . For example, for , let
Thus by the Cantor–Bernstein–Schroeder theorem (or, technically, the dual version of the theorem), and have the same cardinality.