Sequences with 0's in $\mathbb R ^\omega$ Let $\mathbb R ^\omega$ be the set of all sequences of real numbers in the product topology.
Let $X$ be the set of all sequences in $\mathbb R ^\omega$ which have at least one 0.
Let $Y$ be the set of all sequences in $\mathbb R ^\omega$ which have at least two 0's.
Are $X$ and $Y$ homeomorphic, and if so, is there a simple proof of this?
 A: For every $n\in\mathbb N$ consider the space
$$X_n=\{x\in\mathbb R^\omega:\lvert x^{-1}(0)\rvert\ge n\}.$$
Theorem. For any positive integer numbers $n<m$, the spaces $X_n$ and $X_m$ are not homeomorphic.
Proof. The spaces $X_n$ and $X_m$ are not homeomorphic because the space $X_m$ is a $\sigma Z_{m-1}$-space whereas $X_n$ is not.
Let us recall that a closed subset $A$ of a topological space $X$ is a $Z_k$-set in $X$ if the set $C([0,1]^k,X\setminus A)$ is dense in the function space $C([0,1]^k,X)$ endowed with the compact-open topology. A topological space $X$ is called a $\sigma Z_k$-space if $X$ is the countable union of $Z_k$-sets.
$Z_k$-sets are higher-dimensional counterparts of nowhere dense sets and $\sigma Z_k$ are counterparts of meager spaces.
Since any singleton in $\mathbb R^m$ is a $Z_{m-1}$-set, the space
$X_m$ is a $\sigma Z_{m-1}$-space. Assuming that $X_m$ is homeomorphic to $X_n$, we would conclude that $X_n$ is a $\sigma Z_{m-1}$-space. Consider the closed subset $A=\{0\}^n\times\mathbb R^{\omega\setminus n}\subseteq X_n$ of $\mathbb R^\omega$ and observe that $A$ is not a $Z_n$-set in $\mathbb R^\omega$.  Since $X_n$ is a $\sigma Z_{m-1}$-space, $A=\bigcup_{i\in\omega}A_i$ is a countable union of $Z_{m-1}$-sets $A_i$. Since $n\le m-1$, each $Z_{m-1}$-set $A_i$ is a $Z_n$-set and hence the set $C([0,1]^n,X_n\setminus A_i)$ is dense in $C([0,1]^n,X_n)$.
It can be shown that the set $C([0,1]^n,X_n)$ is dense in $C([0,1]^n,\mathbb R^\omega)$ and hence $C([0,1]^n,X_n\setminus A_i)$ is dense in $C([0,1]^n,\mathbb R^\omega)$, which means that $A_i$ is a $Z_n$-set in $\mathbb R^\omega$ (being a closed subset of the closed subset $A$ of $\mathbb R^\omega$). By the proof of Lemma 2.7 in the paper Concerning locally homotopy negligible sets and characterization of $l_2$-manifolds of Toruńczyk, the completeness of $\mathbb R^\omega$ implies that the closed $\sigma Z_n$-set $A=\bigcup_{i\in\omega}A_i$ is a $Z_n$-set in $\mathbb R^\omega$, which is not true. $\quad\square$
