I have never seen this notation before (graduated with a math degree a few months ago; not in school currently). Here's what I gather from Munkres' Topology:
Given a set $X$, an $\mathbf{\omega}$ -tuple of elements of $X$ is a function \begin{equation*} \mathbf{x}: \mathbb{N} \to X\text{.} \end{equation*} Such a function is called a sequence or infinite sequence of elements of $X$. The $i$th value of $\mathbf{x}$ at $i$ is $x_i$, known as the $i$th coordinate of $\mathbf{x}$ and denote \begin{equation*} \mathbf{x} = \left(x_1, x_2, \dots\right) = \left(x_n\right)_{n \in \mathbb{N}}\text{.} \end{equation*}
As an example, Munkres mentions $X^{\omega}$, where $X = \{0, 1\}$. What is $X^{\omega}$ in this case? The set of all $\omega$-tuples where each coordinate can either be a $0$ or $1$? So am I correct in thinking that $$X^{\omega} = \{(x_1, \dots, x_{\omega}): x_i = 0 \text{ or }1\}\text{?}$$ Can $\omega$ be finite, as well as infinite?