Question 7 in Munkres's Topology Reads:
"Let $\mathbb{R}^\infty$ be a subset of $\mathbb{R}^\omega$ consisting of all sequences that are 'eventually zero...'"
I am trying to understand what's the difference between $\mathbb{R}^\infty$ and $\mathbb{R}^\omega$. What does the superscript $\omega$ suggest about the space $\mathbb{R}^\omega$ ?
I believe $\mathbb{R}^\infty$ refers to the product topology (or Box Topology) of $\mathbb{R}$, while $\mathbb{R}^\omega$ $\textit{is}$ $\mathbb{R}^\infty$, but also holds the property that any sequence is eventually zero...
Is this correct? I have not seen the formal definition of either of these and in this case, I believe the definition directly leads me to answer the problem: " What is the closure of $\mathbb{R}^\infty$ in $\mathbb{R}^\omega$ in the box and product topologies?"
$\mathbb{R}^\omega$ is the product of denumberably many
copies of R with the product topology,
namely all sequences of real numbers.
$\mathbb{R}^\infty$ is just the sequences that are eventually 0.
Between these two spaces is the space of real
sequences that converge to zero.