I want to know which is the topology of $\mathbb{R}^{\infty}$, but I don't know even how to start to give it one. How can I give to $\mathbb{R}^{\infty}$ a topology? Is there a book that explains this?
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I guess the book Topology by J.Munkres should explain in more detail this space. – Surb Sep 20 '18 at 15:00
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Or does $\mathbb{R}^\infty$ refer to the Alexandroff compactification (or one-point-compactification) $\mathbb{R}\cup{\infty}$? Then the topology of $\mathbb{R}^\infty$ would be given by the topology of $\mathbb{R}$ plus all sets $\mathbb{R}-C\cup {\infty}$ where $C\subset\mathbb{R}$ is conmpact and closed in $\mathbb{R}$. – FWE Sep 20 '18 at 15:09
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@Surb That question asks for a definition of $\Bbb{R}^\infty$ in the special case we want to give it the structure of a CW-complex. While it is very likely that this question has been asked and answered earlier I'm not sure that's the right dupe target. – Jyrki Lahtonen Sep 20 '18 at 20:06
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There are two standard topologies on $\mathbb{R}^\infty$. (You have to be careful with the notion $\infty$ I assume it means countable infinity so just write $\mathbb{R}^\mathbb{N}$).
The first topology is called the product topology and is the more common than the other one. One can define this topology by forcing that $x_n\rightarrow x$ if and only if every coordinate of $x_n$ converges to the corresponding coordinate of $x$ (i.e. point-wise convergence).
The second topology is called the box topology. This topology is less common. One can define this topology by forcing that $x_n\rightarrow x$ if each coordinate converges to $x$ uniformly (i.e. they convergence pointwise but the big $N$ is only dependent on the $\varepsilon$ and is not dependent on the coordinate).
Yanko
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