# Dispensing with the notion of infinity for the sake of coverings [closed]

Instead of taking a one to one correspondence meaning each set has the same number of elements. why not use the concept of coverings of topology? The irrational numbers covers the whole numbers but not vice versa?

A hierarchy of coverings instead of infinities. Wouldn't that make those infinities more manageable in those terms?( yes I know topology can be expressed in set theory)

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## closed as not a real question by Reid Barton, Tom Leinster, Ben Webster♦Mar 25 '10 at 14:54

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This is in danger of being closed as "not a real question". As the answers below testify, at present it is simply unclear what you mean. Please clarify... – Pete L. Clark Mar 25 '10 at 14:54
In particular, why do you want to "dispense with the notion of infinity"? What is unmanageable about the current notions of infinity? – Pete L. Clark Mar 25 '10 at 14:56

By $X$ covers $Y$ I assume you mean there exists a surjection $f:X \to Y$. The theory you're describing is exactly the same as the standard theory of cardinal numbers. In fact, if $X$ "covers" $Y$ and $Y$ "covers" $X$ then there is a bijection between $X$ and $Y$ . The proof is pretty and easy and is a good homework problem. You could also look it up in the beginning of any book that introduces the cardinals. Aside: I don't see what this has to do with topology. I also don't understand what you mean by "Dispensing with the notion of infinity..."