A three-dimensional Euclidean space is a special case of a Euclidean space. In terms of Bourbaki, the species of three-dimensional Euclidean space is richer than the species of Euclidean space. Likewise, the species of compact topological space is richer than the species of topological space.
https://en.wikipedia.org/wiki/Space_(mathematics)
mathematical space - (mathematics) any set of points that satisfy a set of postulates of some kind; assume that the topological space is finite dimensional.
http://www.thefreedictionary.com/mathematical+space
Points and lines are two of the most fundamental concepts in Geometry, but they are also the most difficult to define. We can describe intuitively their characteristics, but there is no set definition for them: they, along with the plane, are the undefined terms of geometry.
https://en.wikibooks.org/wiki/Geometry/Points,_Lines,_Line_Segments_and_Rays
Atoms are made of 99.99999% space, so it turns out that what we call reality is mostly space with a little bit of a jiggle—a little vibratory fluctuation or, as described in quantum theory, a waveform generating what we call atomic structure.
http://www.planetpuna.com/haramein-seminar/THE%20GEOMETRY%20OF%20SPACE.pdf
In the ancient mathematics, space was a geometric abstraction of the three-dimensional space observed in the everyday life. Axiomatic method had been the main research tool since Euclid (about 300 BC). The method of coordinates (analytic geometry) was adopted by Rene René descartes in. 1637
https://en.wikipedia.org/wiki/Space_(mathematics)
In mathematics, a space is a set (sometimes called a universe) with some added structure. Mathematical spaces often form a hierarchy, i.e., one space may inherit all the characteristics of a parent space.
https://en.wikipedia.org/wiki/Space_(mathematics)
In this case water can be represented as atomic structure and ice as the energy of the vacuum structure. Now we can identify both sides of the equation--one being a wave form (water or liquid) and the other being geometric (solid) as it gets cooler. This research identifies the geometric relationship between space and the phase experienced by matter as related to [or illustrated by] the water/ice example.
http://www.planetpuna.com/haramein-seminar/THE%20GEOMETRY%20OF%20SPACE.pdf
The vector directions are at a 180 phase from each other or, in other words, in opposite directions, whereby the energy dispensed by your body is from the boundary outside of yourself and the information gatheri ng is from the boundary inside of yourself, i.e. Consciousness or self-awareness.
http://www.planetpuna.com/haramein-seminar/THE%20GEOMETRY%20OF%20SPACE.pdf
The relationship is geometrically communicated as a fractal tetrahedral array circumscribed by a sphere where the vector lines of forces generated by the geometric array is the structure of space itself, and the sphere is the waveform resulting from the fluctuation of space, that we observe as atoms.
http://www.planetpuna.com/haramein-seminar/THE%20GEOMETRY%20OF%20SPACE.pdf
That point is called the end-point of the ray. Note that a line segment has two end-points, a ray one, and a line none. An angle can be formed when two rays meet at a common point. The rays are the sides of the angle.
https://en.wikibooks.org/wiki/Geometry/Points,_Lines,_Line_Segments_and_Rays
definition of a space in geometry
Space was a geometric abstraction of the three-dimensional space observed in the everyday life.