Physlib.SpaceAndTime.Time.TimeMan

The manifold $\text{TimeMan}$ is equipped with a topological space structure. This topology is the induced topology (the coarsest topology) such that the map $\text{val} : \text{TimeMan} \to \mathbb{R}$ is continuous, where $\mathbb{R}$ carries its standard topology.

The map $\text{val} : \text{TimeMan} \to \mathbb{R}$, which relates the time manifold $\text{TimeMan}$ to the real numbers $\mathbb{R}$, is surjective.

The range of the map $\text{val} : \text{TimeMan} \to \mathbb{R}$, which maps elements of the time manifold to the real numbers, is the set of all real numbers $\mathbb{R}$.

The map $\text{val} : \text{TimeMan} \to \mathbb{R}$ is an inducing map, meaning that the topology on the time manifold $\text{TimeMan}$ is the induced topology (the pullback topology) from the standard topology on $\mathbb{R}$ via the map $\text{val}$.

The map $\text{val} : \text{TimeMan} \to \mathbb{R}$, which maps elements of the time manifold to the real numbers, is injective.

The map $\text{val} : \text{TimeMan} \to \mathbb{R}$, which maps elements of the time manifold to the real numbers, is an open embedding.

For any subset $s \subseteq \text{TimeMan}$, $s$ is an open set in the time manifold if and only if its image $\text{val}(s) = \{ \text{val}(x) \mid x \in s \}$ is an open set in the real numbers $\mathbb{R}$ under the standard topology.

The map $\text{val}: \text{TimeMan} \to \mathbb{R}$ is a homeomorphism between the time manifold $\text{TimeMan}$ and the real numbers $\mathbb{R}$. It identifies elements of $\text{TimeMan}$ with their corresponding real values, where $\text{TimeMan}$ is equipped with the topology induced by the $\text{val}$ map.

The time manifold $\text{TimeMan}$ is equipped with the structure of a charted space modeled on the real numbers $\mathbb{R}$. This structure is defined by an atlas containing a single global chart, which is the homeomorphism $\text{val} : \text{TimeMan} \to \mathbb{R}$ that identifies points in the time manifold with their corresponding real values.

The time manifold $\text{TimeMan}$ is a real-analytic ($C^\omega$) manifold modeled on the real numbers $\mathbb{R}$ with the standard model with corners $\mathcal{I}(\mathbb{R}, \mathbb{R})$. This manifold structure is induced by the map $\text{val} : \text{TimeMan} \to \mathbb{R}$, which identifies the manifold with the real line.

The map $\text{val} : \text{TimeMan} \to \mathbb{R}$ is real-analytic (of class $C^\omega$) as a map between manifolds, where both the time manifold $\text{TimeMan}$ and the real numbers $\mathbb{R}$ are equipped with their standard manifold structures modeled on $\mathbb{R}$ with the model with corners $\mathcal{I}(\mathbb{R}, \mathbb{R})$.

The map $\text{val} : \text{TimeMan} \to \mathbb{R}$ is a real-analytic ($C^\omega$) diffeomorphism between the time manifold $\text{TimeMan}$ and the real numbers $\mathbb{R}$. This diffeomorphism is established using the homeomorphism $\text{val}$ and the fact that both $\text{val}$ and its inverse are of class $C^\omega$ with respect to the standard manifold structures on $\text{TimeMan}$ and $\mathbb{R}$.

This definition establishes a "less than or equal to" relation $\le$ on the time manifold $\text{TimeMan}$. For any two points $x, y \in \text{TimeMan}$, the relation $x \le y$ holds if and only if their corresponding real values satisfy $x.\text{val} \le y.\text{val}$, where $\text{val}: \text{TimeMan} \to \mathbb{R}$ is the canonical map identifying the manifold with the real numbers. This ordering provides a way to define an orientation on the manifold.