nLab
continuum

Idea

A continuum is in general something opposite to a discrete. There are several notions of continuum in mathematics:

• often the continuum is taken to be the real line; consequently the cardinality of the continuum is the cardinality of the set of real numbers. There is a related continuum hypothesis? in set theory.

• a metric continuum is any compact connected metric space

In physics, by a continuum one usually means a medium which spreads the physical quantities spatially with some finite density, unlike the physics of a system of particles, where an infinite (delta/function like) density is attached to a discrete system of points.

In cohesive homotopy type theory

One can try to axiomatize the notion of continuum in cohesive homotopy type theory. There the idea of an object ${𝔸}^1$ all whose points are, while different, connectable by continuous paths is encoded by asking that after applying the fundamental ∞-groupoid functor $\Pi$ to it, the result is something contractible

$$\Pi ({𝔸}^1)\simeq *.$$\mathbf{\Pi}(\mathbb{A}^1) \simeq * \,.

For instance in the model of homotopy cohesion called Smooth∞Grpd we have a full and faithful embedding of smooth manifolds. Therefore we can embed the integers $\mathbb{Z}$, the rational numbers $ℝ$ as well as the real numbers $ℝ$, all equipped with their canonical smooth manifold structure. This is discrete for the first two, but not for the last one, and homotopy cohesion can detect this:

$$\Pi (\mathbb{Z})\simeq \mathbb{Z};$$\mathbf{\Pi}(\mathbb{Z}) \simeq \mathbb{Z} \,;

$$\Pi (\mathbb{Q})\simeq \mathbb{Q};$$\mathbf{\Pi}(\mathbb{Q}) \simeq \mathbb{Q} \,;

but

$$\Pi (ℝ)\simeq *.$$\mathbf{\Pi}(\mathbb{R}) \simeq * \,.

This reflects the fact that the points of $ℝ$ form a continuum, but those of $\mathbb{Z}$ and $\mathbb{Q}$ do not.

Also the complex numbers $ℂ$ with their canonical manifold structure of course form a continuum

$$\Pi (ℂ)\simeq *.$$\mathbf{\Pi}(\mathbb{C}) \simeq * \,.