Andrey Hihlovskiy

Professional blog on groovy, gradle, Java, Javascript and other stuff.

What is the cardinality of the Euclidean topology?

David Richeson: Division by Zero

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…

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