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 is also a kind of; for instance, a normed vector space is also a metric space. https://en.wikipedia.org/wiki/Space_(mathematics)

Real Projective Space

• 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).

Examples

• 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).