> For the complete documentation index, see [llms.txt](https://wiki.hanzheteng.com/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://wiki.hanzheteng.com/math/topology/topological-spaces.md). # 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 ![{\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). * 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). ### --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter: ``` GET https://wiki.hanzheteng.com/math/topology/topological-spaces.md?ask=&goal= ``` `ask` is the immediate question: it should be specific, self-contained, and written in natural language. `goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.