The extension and abstraction of physical space concepts such as Euclidean space, hyperbolic space, Riemannian space, various function spaces and topological space reflect the development of people's understanding of various attributes of spatial structure.

The earliest concept of mathematical space is Euclidean space. It comes from the intuition of space, reflecting the initial understanding of the linearity, uniformity, isotropy, inclusiveness, positional relationship (distance), three-dimensional, and even infinite extensibility, infinite separability and continuity of space. However, for a long time, people's understanding of space is limited to the scope of Euclidean geometry, and they think that it has nothing to do with time. The appearance of non-Euclidean geometry of 50100.00000000105 breaks through the traditional concept that Euclidean space is the only mathematical space. The space concept of non-Euclidean geometry is more abstract, and it is a special form of Riemannian space, which is unified with Euclidean space into a constant curvature space. /kloc-In the mid-9th century, G.F.B Riemann also introduced the concept of manifold. These concepts not only play a great role in understanding physical space, but also greatly enrich the concepts of space in mathematics.

1At the end of the 9th century and the beginning of the 20th century, people gave the topological definition of dimension, and made an in-depth study on the metric properties of function space, which resulted in a series of important concepts of mathematical space, especially the general concept of topological space. Since 1930s, all kinds of spaces in mathematics have been unified on the basis of mathematical structure, and people have a better understanding of all kinds of mathematical spaces. With the deepening of the understanding of physical space and the development of mathematical research, various mathematical concepts of space have been popularized from algebra, geometry and topology. In algebra, the popularization of the concept of space mainly comes from the emergence and development of analytic geometry. Geometric objects (points, lines, etc.). ) and the array form a corresponding relationship, so that people can accurately and quantitatively describe the space. In this way, it is easy to generalize the coordinate three array to the coordinate n array (vector). Its corresponding space is n-dimensional linear space or vector space. This space expands Euclidean space in dimension, but omits the concept of distance in Euclidean space. The linear space on real number field can usually be extended to general field, especially the linear space on finite field becomes a space with only a limited number of points, and its spatial continuity is abandoned. Algebraically and geometrically, space can be extended to affine space and projective space. Projective space can contain infinite points and infinite lines by geometric method or coordinate method. In addition, various spaces can be made into intuitive models for physics and even other sciences to deal with motion with arrays, phase spaces and state spaces.

The more abstract form of space is topological space. Because the topological structure reflects the relationship between points, the concepts of Euclidean space distance and vector length of vector space are abandoned in topological space.

People's research on various mathematical spaces reflects the process from local and superficial intuition to a deeper understanding of various attributes of space. For example, with the development of topology, people have a deeper and more essential understanding of the dimension, continuity, opening and closing, boundedness and boundedness of space and directionality of space. The study of manifold also makes people understand the finiteness and infinity of space by leaps and bounds, and the concept of manifold is an important development of the concept of space. It is a Euclidean space in part, but it can have various forms as a whole. It can be opened and closed. There can be edges or no boundaries. This profound understanding can promote the study of physical space. For example, Minkowski space is the mathematical model of special relativity, while Riemann space becomes the mathematical model of general relativity (see relativity).