Let $\mathbb R^\infty$ be the subset of $\mathbb R^\omega$ consisting of all sequences that are eventually zero that is all sequences $(x_1,x_2,\ldots)$ such that $x_i\neq 0$ for only finitely many values of $i$. What is the closure of $\mathbb R^\infty$ in $\mathbb R^\omega$ in the box and product topology?
How do you even start to think about this problem? It seems like it is one of the problems from Munkres (but not sure).
Best Answer
Hence $\mathbb{R}^{\infty}$ is dense in $\mathbb{R}^{\omega}$ in the product topology.
Hence $\mathbb{R}^{\infty}$ is closed in the box topology.