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Every space treated in Section "Types of spaces" above, except for "Non-commutative geometry", "Schemes" and "Topoi" subsections, is a set (the "principal base set" of the structure, according to Bourbaki) endowed with some additional structure; elements of the base set are usually called "points" of this space. In contrast, elements of (the ...
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements , it was the three-dimensional space of Euclidean geometry , but in modern mathematics there are Euclidean spaces of any positive integer dimension n , which are called Euclidean n -spaces when one wants to specify their ...
A regular space is necessarily also preregular, i.e., any two topologically distinguishable points can be separated by neighbourhoods. Since a Hausdorff space is the same as a preregular T 0 space, a regular space which is also T 0 must be Hausdorff (and thus T 3). In fact, a regular Hausdorff space satisfies the slightly stronger condition T 2½.
A space M is a fine moduli space for the functor F if M represents F, i.e., there is a natural isomorphism τ : F → Hom(−, M), where Hom(−, M) is the functor of points. This implies that M carries a universal family; this family is the family on M corresponding to the identity map 1 M ∊ Hom(M, M).
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms ...
An example of negatively curved space is hyperbolic geometry (see also: non-positive curvature). A space or space-time with zero curvature is called flat. For example, Euclidean space is an example of a flat space, and Minkowski space is an example of a flat spacetime. There are other examples of flat geometries in both settings, though.
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Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like the Timaeus of Plato, or Socrates in his reflections on what the Greeks called khôra (i.e. "space"), or in the Physics of Aristotle (Book IV, Delta) in the definition of topos (i.e. place), or in the later "geometrical conception of place" as "space qua extension" in the ...
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