Countable-dimensional sphere

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Definition

As the sphere in countable-dimensional real vector space

Denote by $\mathbb {R} ^{\omega }$ the space of sequences of real numbers (i.e., things of the form $(x_{1},x_{2},\dots ,x_{n},\dots )$) with the property that at most finitely many of the numbers are nonzero. Denote by $S^{\infty }$ the subset given by:

$S^{\omega }=\{(x_{1},x_{2},\dots ,x_{n},\dots )\in \mathbb {R} ^{\infty }\mid \sum _{i=1}^{\infty }x_{i}^{2}=1\}$

Note that the actual summation involves only finitely many nonzero terms, so it is not in fact an infinite sum.

This space $S^{\omega }$, also denoted $S^{\infty }$, is termed the countable-dimensional sphere or infinite-dimensional sphere. What's the topology?

As an inductive limit of finite-dimensional spheres

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Properties

• Countable-dimensional sphere is contractible