Space

A space is a continuum of points. The dimensionality of a space is the amount of coordinates required to specify the location of a point (excluding spacefilling curves and the like). Spaces can be embedded in higher-dimensional spaces. For example, in 3-dimensional space, a plane is an embedding of 2-dimensional euclidean space while a sphere is an embedding of 2-dimensional spherical space.

Curvature
Spaces can be divided into three categories based on curvature:

Spherical
Spherical space is finite and has positive curvature everywhere. The circumference of a circle is always less than 2πr. The angles in a triangle add up to more than 180°. The sum of the squares of the legs of a right triangle is always less than the square of the hypotenuse.

Euclidean
Euclidean space is infinite and has zero curvature everywhere. The circumference of a circle is equal to 2πr. The angles in a triangle add up to exactly 180°, a direct consequence of the parallel postulate. and the sum of the squares of the legs of a right triangle is always equal to the square of the hypotenuse; in other words, the Pythagoream Theorem holds.

Hyperbolic
Hyperbolic space is infinite and has negative curvature everywhere. The circumference of a circle is always greater than 2πr. The angles in a triangle add up to less than 180°. The sum of the squares of the legs of a right triangle is always greater than the square of the hypotenuse.