Co-finite topology on an infinite set

[Deveno]

If τ is the co-finite topology on an infinite set X, does there exist an injection from τ to X? I'm having trouble wrapping my mind around this.

on the one hand, for A in τ, we have A = X - S, for some finite set S. so it seems that there is a 1-1 correspondence:

A <--> S, of τ with the finite subsets of X.

so if X were countable, it seems that the set of all finite subsets would also be countable (i could put N into a 1-1 correspondence with the algebraic numbers, for example, and what is an algebraic number but it's associated minimal polynomial, and what is an integral polynomial except a finite sequence of integers (its coefficients)?).

but if X is uncountably infinite, i don't know if τ is "bigger" than X (it's certainly at least as big). certainly τ is uncountable (since it contains X - {x} for every element x of X). i suspect that it is not, that if we "group" the co-finite sets by the size of their complements, then |τ| = |N|*|X| < |X x X| = |X|. is this true?

 

[morphism]

i suspect that it is not, that if we "group" the co-finite sets by the size of their complements, then |τ| = |N|*|X| < |X x X| = |X|. is this true?

Yup. Coupled with your observation about complements of singletons, this proves that |τ|=|X| if X is infinite.

 

[Amir Livne]

To complete the classification, you can see that if X is finite and has n elements, all of its subsets are finite, and there are 2ⁿ of them, which is more than n. So there is never a 1-1 correspondence between τ and X in the finite case.