Space (geometry) | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-03

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Space_(mathematics)


00:01:35 1 History
00:01:43 1.1 Before the golden age of geometry
00:05:14 1.2 The golden age and afterwards
00:08:35 2 Taxonomy of spaces
00:08:45 2.1 Three taxonomic ranks
00:12:44 2.2 Relations between species of spaces
00:17:43 3 Types of spaces
00:17:52 3.1 Linear and topological spaces
00:22:16 3.2 Affine and projective spaces
00:24:40 3.3 Metric and uniform spaces
00:26:32 3.4 Normed, Banach, inner product, and Hilbert spaces
00:29:11 3.5 Smooth and Riemannian manifolds
00:30:42 3.6 Measurable, measure, and probability spaces
00:34:48 3.7 Non-commutative geometry
00:40:11 3.8 Schemes
00:45:48 3.9 Topoi
00:49:11 4 Spaces and structures
00:53:12 5 Mathematical spaces by name
00:53:22 6 See also
00:53:31 7 Notes



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SUMMARY
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In mathematics, a space is a set (sometimes called a universe) with some added structure.
While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. 
The nature of the points can vary widely: for example, the points can be elements of a set, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space. More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships between the points of a three-dimensional Euclidean space are uniquely determined by Euclid's axioms, and all three-dimensional Euclidean spaces are considered identical.
Topological notions such as continuity have natural definitions in every Euclidean space. 
However, topology does not distinguish straight lines from curved lines, and the relation between Euclidean and topological spaces is thus "forgetful". Relations of this kind are sketched in Figure 1, and treated in more detail in the Section "Types of spaces". 
It is not always clear whether a given mathematical object should be considered as a geometric "space", or an  algebraic "structure". A general definition of "structure", proposed by Bourbaki, embraces all common types of spaces, provides a general definition of isomorphism, and justifies the transfer of properties between isomorphic structures.