# Topological Spaces

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

<figure><img src="https://3310611219-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-LwRt8_BWYScDaMKj3wZ%2Fuploads%2FPFqDDngLOtFsUdo6cfxM%2Fimage.png?alt=media&#x26;token=764993bf-3d0b-45f5-89ff-85a9c83db04c" alt="" width="563"><figcaption><p>Overview of types of abstract spaces. (Figure taken from Wikipedia.)<br>An arrow indicates <em><strong>is also a kind of</strong></em>; for instance, a normed vector space is also a metric space.<br><a href="https://en.wikipedia.org/wiki/Space_(mathematics)">https://en.wikipedia.org/wiki/Space_(mathematics)</a></p></figcaption></figure>

## Real Projective Space

* RP^*n* or ![{\displaystyle \mathbb {P} \_{n}(\mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3642da8d0933c939b1d3919e7aad44cb0476fa2d), is the topological space of lines passing through the origin 0 in R^(*n*+1).&#x20;
* 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^*&#x33; → 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).

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