space - what are dimensions?

First, discrete examples.

In a computer screen I can specify any "2D" point with just one single number (pixel order starting count from first at upper left, and going on, left2right and up2down like reading till last one at right bottom corner)

Then I don't need two numbers to specify position, just one.

Same can extend to other "dimensions" I can use same trick to describe a point inside a discrete cube, once have covered a screen like "area" descend a plane level

plane

0 .....5
6 .....10
11.....15

then a cube (made of above planes)

0  ____ 15
16 ____ 31
32 ____ 47

so a single number can map any point in a 3d discrete cube

what about continuous one?

same thing using differential dx instead of discrete points or planes, and that's all

(of course as differential is not well defined as discrete this won't be so easy)

Anyway my question is:

what is a good definition of dimension that avoid these tricks? there is one?

Finally another way to take 3d to 1D, is a reversible transformation(Pairing function) like this

3d point = (x,y,z)

W = x + y * 2^20000 + z * 2^40000

the only "restriction" for this method is that equivalent 3d coords should be lesser than 2^20000, a very easy thing, at least in the known universe even in Plank's units

(20000 and 40000 could of course be changed for lesser numbers, but I like concrete examples)

Answer

It's true that if you just have a set of points, with no additional mathematical structure, the notion of "dimension" is problematic as you say. But the spaces we deal with in physics usually have extra structures that make the notion well-defined. Often, the definition works by making precise a notion of different "directions" at a given point, and then finding a way of counting how many of those directions are independent of each other.

For instance, we often work with vector spaces (in which it makes sense to talk about adding vectors, etc.). The dimension of a vector space is well-defined: it's the maximum number of independent vectors you can find (such that no linear combination of them add to zero).

Also, we often talk about geometrical spaces such as differentiable manifolds, in which there's a notion of "smoothness" of some sort. Once again, these manifolds have a well-defined notion of dimension, and once again it's essentially the number of independent directions you can identify at any given point. In fact, for a smooth manifold one way to define the dimension is to note that you can smoothly map a subsection of the manifold onto a vector space, and then figure out the dimension of the vector space as before.