Bob, You are equating this mathematical 2D space with a 2D plane in a 3D space.
From the en.wikipedia.org
A purely mathematical 2D space has go no connection whatsoever with areas or geometry.
From the en.wikipedia.org
We cannot assume that a purely mathematical 2D space has got any connection with a 2D plane in a 3D space.
From the en.wikipedia.org
A mathematical space exists whenever we have a set of observations and quantitative measures of their attributes.
From the en.wikipedia.org
You seem to be assuming that a mathematical 2D space can be represented by a 2D plane as we understand such in a 3D space.
From the en.wikipedia.org
This technique, called skin texture analysis, turns the unique lines, patterns, and spots apparent in a person's skin into a mathematical space.
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A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated.
From the en.wikipedia.org
Bob, You gave a plane geometrical interpretation to the mathematical concept of a 2D space.
From the en.wikipedia.org
Thus, for the time being at least, these inkblots of space are mere mathematical figments.