What is $X^{\omega}$ where $X$ is a set?

Question

I fail to find a duplicate. If it exists, please link me in the comments and I will delete the question.

In my recently bought topology book, they use $X^{\omega}$ where $X$ is a set. However, this is never defined in the book, probably because it is viewed as a prerequisite (that I obviously lack). Can anyone explain what this means? I skimmed the Wikipedia-article on ordinals, without getting any wiser.

Answer 1

Think of $ω$ as the natural numbers (one uses $ω$ when talking about the natural numbers as an ordinal number).

Say $X$ is a set. Then $X^ω$ is defined as the set of all functions from $ω$ to $X$: $$X^ω:=\{f:ω\rightarrow X\}$$

Each function $f\in X^ω$ (or lets use $x$ instead of $f$ - so) $x\in X^ω$ is determined by its image in $X$ - so: $$X^ω=\{(x(0),x(1),..)|x\in X^ω\}$$

This is the same as thinking of $X^ω$ as the set of infinite sequences of elements of $X$: $$X^ω=\{(x_0,x_1,..)|x_i\in X,\space i=0,1,..\}$$

Answer 2

If $X$ and $A$ are sets then $X^A$ denotes the set of all functions $A\rightarrow X$. Here $A=\omega=\{0,1,2,\dots\}$.

Answer 3

In theoretical computer science, $X^\omega$ usually denotes the set of infinite words on the alphabet $X$.