What is $\mathbb R^\omega$?
I have seen $\mathbb R^\omega$ mentioned in my topology texts but cannot find where $\omega$ is defined. Could someone please tell me what it means in comparison to $\mathbb R^n$?
$\endgroup$ 22 Answers
$\begingroup$tl; dr;
$$X^\omega = X\times X\times\cdots$$
is an infinite countable product of $X$, i.e. the set of all countable sequences over $X$.
Longer explanation: so $\omega$ is an ordinal number defined as the ordinal number of naturals $\mathbb{N}$. If we use the Von Neumann definition of ordinals then for an ordinal number $k$ we have a very straight forward generalization of finite Cartesian product:
$$X^k=\{f:k\to X\ |\ f\text{ is a function}\}$$
and so you can think about $X^k$ as the set of all ordered "sequences" with values in $X$. For finite $n$ this coincides with classical finite product because $n$ as a Von Neumann ordinal is recursively defined as $\{0,1,2,\ldots,n-1\}$ with $0=\emptyset$. For $\omega$ (which in Von Neumann case is simply $\mathbb{N}$) this coincides with the usual notion of sequences. For uncountable ordinals this probably goes beyond any intuition. Nevertheless typically by $X^\omega$ people understand the countable infinite product $X\times X\times\cdots$.
Note that there are alternative (and arguably simplier) definitions of an infinite Cartesian product, e.g. the set of all choice functions.
$\endgroup$ $\begingroup$The cardinality of $\Bbb N$, which is usually written $\aleph_0$ (the smallest infinite cardinal), is sometimes/often also written $\omega$. In other words
$$|\Bbb N| = \aleph_0 = \omega$$
Hence $\Bbb R^{\omega} = \Bbb R \times \Bbb R \times \Bbb R \times \dots$ is the cartesian product of countably infinitely many $\Bbb R$.
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