What are the components and path components of $\mathbb{R}^\omega$ (in the product topology)?
At the moment, I am just working on the first part of the question. I read somewhere online that a space $X$ has only one component if and only if $X$ is connected. Since $\mathbb{R}^\omega$ is connected, then according to this fact $\mathbb{R}^\omega$ is the only component. I suspect the same is true of path-connectedness, but hopefully someone will kindly correct me if this is wrong.
Unfortunately, this fact about connectedness hasn't been presented in the book I am working through. How could one solve this without that fact; I must confess, I have really no intuition about components yet, especially in the infinite dimensional space $R^ω$. I could use some hints.
As a way of knowing what ideas/theorems I have available to me, this is problem 2a in section 25 of chapter 3 in Munkres' Topology.
EDIT:
Theorem 25.1: The components of $X$ are connected disjoint subspaces of $X$ whose union is $X$, such that each nonempty connected subspace of $X$ intersects only one of them.
I believe this is a proof: If $X$ only has one component, and it is suppose to partition $X$, then every element of $X$ must be in this single component, proving that $X$ equals this single component; moreover, since components are connected, it follows that $X$ is connected.
Now, suppose that $X$ is connected but has at least two components. Since components are nonempty, each has at least one element. But $X$ is a connected subspace containing both of these elements, and it certainly intersects both components, which is at odds with theorem 25.1. Hence, $X$ can only have one component.