What is your definition of "dimension"? Are you familiar with space filling curves, which are continuous, surjective functions from an interval to a square?

@XanderHenderson I've seen them before(a little bit), but the question was inspired by the compactification in string theory, i.e. with a Ricci-flat manifold, which was why I used the phrase "test particle". the space filling curves was not always a geodesic, but the one given in the post was. (which made it a lot of more interesting.)

That still doesn't answer my first question: what do you mean by "dimension"?

@XanderHenderson Could it been seen as a single one dimensional object along the particle's trajectory, or was $(x,y)$ necessary.

What is a "one dimensional object"? What is your definition of dimension?

@XanderHenderson en.wikipedia.org/wiki/One-dimensional_space

That definition gives dimension in terms of vector space structures; essentially, the dimension of a space is the dimension of its tangent space (which is a vector space, which has a natural notion of dimension over its base field).

Or this one: en.wikipedia.org/wiki/Product_topology it's the simple product topology

The torus is topologically two dimensional for any reasonable notion of dimension (e.g. small or large inductive dimension; Lebsgue covering dimension)

It is not, however, a product space, though it can be described as a quotient space with respect to an equivalence relation on $\mathbb{R}^2$, which is a product space.

But as mentioned in the post, both coordinates and measure could be complete by a single test trajectory.

The topology it inherits from this quotient is two-dimensional.

But it could be mapped by an isomorphic map. even open sets to open sets.

Isomorphic in which category?

From $\mathbb{R}^2$ to $\mathbb{R}$

A bijection function.

A bijection is an isomorphism in the category of sets. To the best of my knowledge, there is no useful notion of "dimension" which applies to sets.

The relevant metric there is "cardinality".

Note that the cardinality of a line is, in fact, equal to the cardinality of a square.

In order to reasonably define dimension, additional structure is required. The mapping described by the particle you describe will not preserve those structures.

There doesn't need a point disappear on the trajectory, it's a trajectory. Also it's measure zero and it's not a pole so it doesn't matter(in physics).

(A point was a measure zero.)

My point was that a space filling curve or "trajectory" is not a homeomorphism.

That it has measure zero is irrelevant (and, indeed, meaningless thus far, as I am talking only of the topology of the spaces, and have not introduced measures).

So if you define "dimension" topologically, the torus is 2-dimensional, which doesn't contradict the existence of space filling curves, as such curves don't preserve the topological structure.

So, again, what do you mean by "dimension"? Be very precise.