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I am currently attending a course on point-set topology and fundamental group. Today we proved in class that $\mathbb{R}$ and $\mathbb{R}^2$ are not homeomorphic with their normal Hausdorff topologies. I also learned from this question that there is in fact no continuous bijection between $\mathbb{R}$ and $\mathbb{R}^2$.

I am wondering, what happens when we consider other topologies on $\mathbb{R}$ and $\mathbb{R}^2$? For example, in our course, we discussed cofinite and cocountable topologies, which I denote by $\tau_f$ and $\tau_c$ respectively.

I am wondering, are there continuous bijections between $(\mathbb{R}, \tau_c)$ and $(\mathbb{R}^2, \tau_f)$? What about between $(\mathbb{R}^2, \tau_c)$ and $(\mathbb{R}, \tau_f)$?

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    The cofinite and cocountable topologies reduce it to a question of cardinality. Any injective map between two spaces, both endowed with the cofinite topology, or both endowed with the cocountable topology, is continuous. – Daniel Fischer Mar 01 '16 at 19:33
  • @DanielFischer Yes, I agree. But I am asking if there is a continuous bijection from a cocountable topology to a cofinite topology. – user319101 Mar 01 '16 at 19:35
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    Well, the cocountable topology on $X$ is finer than the cofinite topology on $X$. – Daniel Fischer Mar 01 '16 at 19:36
  • @DanielFischer Ah, I see, the question is trivial, thanks a lot! – user319101 Mar 01 '16 at 19:39

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