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I was reading the "topology" article on Wikipedia and They stated the following:

"For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies."

My question is, which set and what different topologies generate the real line and the complex plane? I don't see how this is possible. Because clearly $\mathbb R \neq \mathbb C$, so how do we have a set $K$ such that $(K,\tau_1) = \mathbb R$, but $(K,\tau_2) = \mathbb C$?

Eduardo Magalhães
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    A guess: because they have the same size, so we have bijections between them (but not bijections that preserve the topology!) – Henno Brandsma Aug 08 '20 at 14:58
  • Can we take as granted that you know that $\mathbb R$ and $\mathbb C$ are the sets of the same size? (Sure $\mathbb C$ "looks" bigger, but it actually isn't, in the same sense as the set of all integers is of the same size as the set of all even integers?) –  Aug 08 '20 at 15:02
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    At the time of this comment, the relevant passage in Wikipedia is: The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies. The second sentence is very misleading. Indeed, without applying a significant literary license, it's just incorrect. The "For instance" should be followed with something like the set of real numbers endowed with different topologies, thus giving different topological spaces on the same set. Someone who can/knows-how-to edit Wikipedia should look at this. – Dave L. Renfro Aug 08 '20 at 15:48
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    @DaveL.Renfro Agreed. That explanation is quite confusing. As sets, the real numbers, the complex numbers, and a Cantor set are "isomorphic" (they have the same cardinality, which means that they are isomorphic in the category of sets). However, put just about any additional structure on them, and they are distinguishable (e.g. $\mathbb{R}$ and $\mathbb{C}$ are different as topological spaces, as fields, as metric spaces, etc). I am not terribly motivated to edit Wikipedia (for fear of wiki warriors), but agree that someone should fix that explanation. – Xander Henderson Aug 08 '20 at 16:46

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The idea here is that these sets have the same cardinality, so there is a bijection of sets $\mathbb{R}\to \mathbb{C}$. So, you might consider these isomorphic as sets.

Of course, the topologies are clearly not the same since if you excise a point from $\Bbb{R}$ with its usual topology it becomes disconnected, while this is not true for $\Bbb{C}$.

As a toy example of this phenomenon: think about the set $\Bbb{N}$ equipped with the discrete topology and also equipped with the finite complement topology (e.g. $U\subseteq \Bbb{N}$ is open if and only if $U^c$ is a finite set). These sets are "identical" but have different topologies.

Alekos Robotis
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