Is $\underline{2}^\mathbb{N} \cong \prod_{n\in\mathbb{N}}\underline{n}$? Let $\mathbb{N}$ denote the set of positive integers, and for $n\in\mathbb{N}$ let $\underline{n}$ be the discrete space on $n$ points.
Is $\underline{2}^\mathbb{N} \cong \prod_{n\in\mathbb{N}}\underline{n}$?
 A: Both are non-empty, compact, second countable, zero-dimensional, Hausdorff spaces without isolated points, so they are homeomorphic by a classical characterization of the Cantor space.
A: My suspicion at the moment is that they are indeed homeomorphic. I'll propose a map from $\prod_{n\in\mathbb{N}}\underline{n}$ to $\underline{2}^\mathbb{N}$ for the community's consideration. It's convenient to denote the elements of $\underline n$ by $\{0,1,\dots,n-1\}$.
For each $n\ge1$, consider the following function $f_n$ from $\underline n$ to the set of finite binary strings:


*

*If $n=1$ then $f_1(0)$ is the empty string.

*If $n=2^k$, then for $a\in\underline n$, define $f_n(a)$ to be the result of writing $a$ as a $k$-bit binary string (with leading 0s if appropriate).

*If $2^k<n<2^{k+1}$, then:

*

*if $a < 2^k$, define $f_n(a)$ to be the result of writing $a$ as a $(k+1)$-bit binary string (necessarily beginning with 0);

*if $2^k \le a < n$, define $f_n(a)$ to be the concatenation of the tiny string 1 with $f_{n-2^k}(a-2^k)$.



For example, when $n=45=(101101)_2$:


*

*the elements $0,\dots,31$ in $\underline{45}$ are mapped to 000000, 000001, ..., 0111111 (in that order);

*the elements $32,\dots,39$ are mapped to 10000, 10001, ..., 10111 in that order (note these strings are shorter);

*the elements $40,41,42,43$ are mapped to 11000, 11001, 11010, 11011;

*the element $44$ is mapped to 111.


Now define $f\colon \prod_{n\in\mathbb{N}}\underline{n} \to \underline{2}^\mathbb{N}$ by letting $f(a_1,a_2,a_3,a_4\dots)$ be the infinite concatenation of $f_2(a_2)$, $f_3(a_3)$, $f_4(a_4)$, .... (Note that we ignore $a_1$ since it always equals $0$.)
I claim that $f$ is at least a set bijection. This follows from the fact that we can recursively recover $a_2$, $a_3$, ... from the image under $f$.
Moreover, I suspect that $f$ is a homeomorphism. The thinking is that, for example, the set of infinite binary strings beginning with 000000 is homeomorphic to the set of infinite binary strings beginning with 111. So somehow the above map, after any finite number of concatenations, has assigned to each element of $\prod_{n\le N} \underline n$ its own cylinder that is still homeomorphic to $\underline2^{\Bbb N}$.
Trying to work out the details hurts my brain, but perhaps others can chime in....
