Topology of $\mathbb{R}^{\infty}$

Question

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?

Answer 1

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).