Countable-dimensional sphere

As the sphere in countable-dimensional real vector space
Denote by $$\R^\omega$$ 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:

$$S^\omega = \{ (x_1,x_2,\dots, x_n, \dots) \in \R^\infty \mid \sum_{i=1}^\infty x_i^2 = 1 \}$$

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?

Properties

 * Countable-dimensional sphere is contractible