• Glome in colors
What is the shape of the universe?
Does it extend infinitely?
We can go around the earth forever but can not find the end. Likewise, the universe may be finite and endless.
Let's consider a mathematical model suitable for such a space.
Any loop or circular path on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture provides that the 3-sphere is the only three-dimensional manifold (up to homeomorphism) with these properties.
• Compact space
Compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded(that is, having all its points lie within some fixed distance of each other).
• Leonard Susskind
The space itself may be more than three dimensions. But we can't visualize more dimensions. The architecture of the brain itself is evolved in the world of three dimensions. We only describe more dimensions by pure mathematics.
In mathematics, a path in a topological space X is a continuous function f from the unit interval I = [0,1] to X
f : I → X.
A topological space for which there exists a path connecting any two points is said to be path-connected.