$\mathbb{R} \times \mathbb{R} \times ... \times \mathbb{R}$ ($n$ times) is homeomorphic to $\mathbb{R}^{n}$

Question

Here is a short sentence from the book "General Topology" of Stephen Willard :

"[...] In particular, $\mathbb{R} \times \mathbb{R} \times ... \times \mathbb{R}$ ($n$ times) with the product topology is homeomorphic to $\mathbb{R}^{n}$."

Recently, I asked a question about the fact that we don't have exactly $\mathbb{R}^{n} = \mathbb{R} \times \mathbb{R} \times ... \times \mathbb{R}$ ($n$ times). Thus, I know now that these two sets are in bijection.

Now, my problem is the following : in the book in question, the author sometimes says things like "the usual topology on $\mathbb{R}^{2}$" and I was wondering :

• If we consider $\mathbb{R}^{n}$, is the "usual topology" not the Tychonoff topology (product topology) ? Which is the same as the ones induced by any norm (considering $\mathbb{R}^{n}$ as a vector space) or as the box topology ?
• If we consider $\mathbb{R} \times \mathbb{R} \times ... \times \mathbb{R}$ ($n$ times), then :
  • Considering that what I said previously is correct (which is maybe not the case), why does the author say that we consider "$\mathbb{R} \times \mathbb{R} \times ... \times \mathbb{R}$ ($n$ times) with the product topology" ?
  • Again, considering that what I said is correct, if we assume that $\mathbb{R}^{n}$ is endowed with its product topology, what is in fact the topology on the set $\mathbb{R} \times \mathbb{R} \times ... \times \mathbb{R}$ ($n$ times) that makes the author says that these two topological spaces are homeomorphic ?

I hope it is clear enough. Thank you for your future answers.

Answer 1

The author is simply making a distinction between the topological product and the set product. By $\mathbb{R}\times\mathbb{R}\times\dotsb\times\mathbb{R}$ he actually means the ( topological) product $(\mathbb{R},\tau_0)\times(\mathbb{R},\tau_0)\times\dotsb\times(\mathbb{R},\tau_0)$ with $\tau_0$ being the order topology on $\mathbb{R}$. On the other hand by $\mathbb{R}^n$ he means the $n$-dimensional topological space i.e. the topology $(\mathbb{R}^n,\tau_1)$ where $\mathbb{R^n}$ is the set product $\mathbb{R}\times\mathbb{R}\times\dotsb\times\mathbb{R}$ and $\tau_1$ is the topology generated by the open balls.

Author's like using these sorts of abbreviations but this can sometimes cause confusion the first time you see it. To avoid this, in fact, I've seen some authors (cannot immediately think of an example) that use $E_n$ to refer to the topological space $(\mathbb{R}^n,\tau_1)$ above.