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In my Real Analysis book, it says a set $S$ is finite if there exists a bijection, $\exists n \in \mathbb{N}$, $f: \{1,2,3,...,n\} \rightarrow S$. Then it says that if a set is not finite, it is said to be infinite.

I was wondering if another way to define an infinite set is if there exists a bijection between another infinite set $S'$ and $S$.

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  • $\begingroup$ A set is indeed infinite if it is "equipotent" with a set otherwise known to be infinite. This is easy to prove by contradiction once you know that a composition of bijections is a bijection. Topology does not really have anything to do with that. $\endgroup$ – Ian Jan 30 '18 at 16:13
  • $\begingroup$ Bijections preserve cardinality, so if you have a bijection between an infinite set a some set $S$ then $S$ is infinite. Topology is irrelevant. $\endgroup$ – Ethan Bolker Jan 30 '18 at 16:13
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this is not a good way to define infinite, but it is a relatively nice way to define different "Sizes" of infinite.

The simplest example, is that a set is said to be "countable" if there exists a bijection $f:\mathbb N \to X$.

You can ignore the "topological" part of the statement, as this has nothing to do with the elements of a topological space $Y$, and in fact, a topological space is a tuple $(Y,\tau)$, where $Y$ is the underlying set, which is the only important thing when construction bijections.

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