Why should I talk about "the properties of space" in Geometry? I have the following statement from Wikipedia's "Geometry" entry:

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

My questions:


*

*If I have an object then I can talk about shape, size, relative position of figures. But what is "the properties of space"?


*Why do we need to talk about "the properties of space" in Geometry?


*What is Space?

 A: The properties of space is given by the system of axioms actually describing a geometry. For example see Hilbert's axioms for the Euclidean geometry. (https://en.wikipedia.org/wiki/Hilbert%27s_axioms)
Now, in this geometry, if you move a triangle ($EFG$) along the straight ($EF$) containing one of the sides as shown below and get the triangle $FKO$

and if you connect $H$ and $O$ with a straight then the new triangle, $HFO$ will be congruent to the former triangles. Whose property is this? Is this a property of the triangle or is this a property of the Euclidean plane (or that of the axiom system describing the Euclidean plane)?
If you still have doubts then consider hyperbolic geometry (https://en.wikipedia.org/wiki/Hyperbolic_geometry) described by its axioms (different from the Euclidean geometry only in the case of the properties of parallels.)
Now, do the same as above (shift a triangle) as shown below in the Poincaré model:

Since the Poincaré model depicts angles like they were Euclidean angles, one can easily tell that the triangle $HFO$ is not congruent with the triangles $EFH$ and $FKO$.
Whose property is that then? Is it that of the triangles or that of space (the axiom system)?
It has to be obvious now that the space (the axiom system describing a geometry) has properties that are not related to the properties of the shapes; so to speak, they are above (meta to) of the properties of the shapes.
To answer the last question "what is space?". The answer is: The space is the axiom system describing the relations of the undefined basic concepts (point, straight line, congruency, etc.)
EDIT
So far, we have been talking about geometry as a branch of mathematics. It is interesting to see what does this branch of mathematics has to do with physical reality. First of all, we have to identify the "undefined concepts" of geometry as some physical objects. Such a basic undefined concept in geometry is the "straights line". One possible interpretation of that concept is "the light beam". Far away from massive objects, as very good approximation: the light beam behaves as if it was an Euclidean object. However, near to large massive and moving objects the light beams do not behave like Euclidean objects. They are bent... When we say "bent", we act like we knew what "unbent" would be like. We do not have any object in the real space compared to which light beams are bent.  The light beams are the "straights" -- by definition. Why do we say then that they are bent? Because: we have a strong desire to see the space as a space with Euclidean features; in which space there are unruly objects like the light beams. But it is impossible to build an universal reference frame in which Euclidean objects exist and to which physical straights could be compared. This is why, doing math is much easier than doing physics.
