Then please explain to me what exactly "space" might be. GR theory only describes distance and 'spacetime'. Space and "time" are not separate entities in GR, they are interrelated based upon the curvatures of spacetime.
Why did time suddenly get scare quotes? Before, you didn't seem to have any trouble with spacetime. Or with time. Which makes your confusion about space rather perplexing.
You're right that space and time are interrelated, though i'm not sure how you can say that since you allegedly don't know what space is.
In Euclidean three-space, we know that rotations mix our three spatial coordinates together. Changing coordinate frames to one that has been rotated around the z axis will change the coordinates of a given point so that its new x and y coordinates are a mixture of the old x and y coordinates.
In Minkowskian four-space, we add a time dimension.
In Special Relativity, changing to a coordinate frame that has a relative velocity compared to the old frame (a
boost), causes not a rotation, but a Lorentz transformation that mixes the space and time coordinates. But we still end up in a coordinate frame with three spatial and one time coordinate.
How do we know? Because the
Minkowski metric still has three positive and one negative entry.
As you say, distance (i.e.
proper length) is defined. And part of that definition is that the time coordinate enters into the formula with an opposite sign to the three spatial coordinates.
In GR, the
metric is more complex, but the same applies. Three coordinates get one sign, and one gets the other. ("the metric is required to be nondegenerate with signature (-+++)") The three coordinates associated with the plus sign are the space coordinates of spacetime, and the remaining coordinate with the negative sign is the time coordinate of spacetime.
Yes, space and time get mixed together, but this does not change the signature of the metric, which still provides for three space and one time coordinate.
So if you're okay with spacetime, the answer to your question is that space (for a particular frame) is the part of spacetime associated with positive entries in the spacetime metric.