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# Topological Spaces

In mathematics, a space is a set with some added structures.

Overview of types of abstract spaces. (Figure taken from Wikipedia.)
An arrow indicates

**; for instance, a normed vector space is also a metric space. https://en.wikipedia.org/wiki/Space_(mathematics)***is also a kind of*- RP^
*n*or , is the topological space of lines passing through the origin 0 in R^(*n*+1). - RP^
*n*can also be formed by identifying antipodal points of the unit*n*-sphere,*Sn*, in R^(*n*+1).

- RP^1 is called the real projective line, which is topologically equivalent to a circle.
- RP^2 is called the real projective plane. This space cannot be embedded in R3.
- RP^3 is (diffeomorphic to) SO(3), hence admits a group structure; the covering map
*S^*3 → RP^3 is a map of groups Spin(3) → SO(3), where Spin(3) is a Lie group that is the universal cover of SO(3).

Last modified 6mo ago