Up to homeomorphism, this space is the unique connected non-compact 1-manifold without boundary, and its one-point compactification gives the homeomorphic to S1.
A group action of this space on the phase space giving the solutions of an ODE is called a flow. Up to homeomorphism, this space is the unique connected non-compact 1-manifold without boundary, and its one-point compactification gives the homeomorphic to S1. In this space bounded monotone sequences converge, implying the (*) Bolzano-Weierstrass theorem. Every non-empty subset of this space has a supremum, and, up to isomorphism, it is the unique Archimedean complete ordered field. The elements of this space can be constructed using Cauchy sequences or as partitions of the rational numbers called Dedekind cuts. Its elements are the fixed points under complex conjugation. For 10 points, name this space composed of the rational and irrational numbers.
A group action of this space on the phase space giving the solutions of an ODE is called a flow.
real numbers [or real line; or R; or R1]