resources: "dimension theory" by hurewicz and wallman, "why space has 3 dimensions" by poincare.
poincare's essay is for the general public on the notion of dimension. he says basically that he calls a finite set zero dimensional for starters. then a set is 1 dimensional if it can be separated by removing a zero dimensional set. e.g. as matt grime pointed out, R^1 is disconnected by the removal of anyone point, hence is one dimensional.
R^2 is not disconnected by removing one point, but is disconnected by removing a copy of R^1 hence R^2 is two dimensional. etc etc..
It has been 40 years since i read it, but i believe hurewicz and wallman defined a topological space to be n dimensional at a point, if there is a basis? of neighborhoods of that point with n-1 dimensional boundaries? so R^1 has dimension one at the origin since there is a local neighborhood basis of intervals, each with boundaries which are just 2 points.
I also like the lebesgue covering dimension definition. It is especially surprizing to me that it is true, and interesting that anyone ever thought of it. it is also useful for defining a special cohomology that is used in algebraic geometry.
ordinary cohomological dimension is based on things being boundaries of other things, except one needs some further conditions, like compactness, since e.g. all R^n spaces for all n, have trivial ordinary cohomology.
The fact that a sphere has dimension 2, is based in this case on the fact that all loops in a sphere are boundaries of discs, but there is a surface, namely the sphere itself, which does not bound anything, within the sphere itself. here one essentially takes some standard objects as representing each dimension, like families of line segments are dimension one, families of triangles are dimension 2, families of tetrahedra are dimension 3, etc etc, and then one maps these standard objects into other spaces.
The lebesgue covering dimension comes into play in the definition of so called cech cohomology, with various coefficients, ultimately "sheaves". Open sets are used to imitate polyhedral structure as follows: A "vertex" is just an open set, an "edge" is a pair of open sets with non empty intersection. A 2 dimensional face is three open sets with non empty intersection. An n dimensional face is a collection of n+1 open sets with non empty interscetion.
Then lebesgue's theorem says that on an n dimensional space a suitably refined open cover will represent an n dimensional polyhedron, since more than n+1 sets will never have an non empty intersection.
Note for example that the structure of an ordinary n dimensional polyhedron is faithfully represented this way by the open cover consisting of the open "stars" of the vertices.
e.g. an interval is recaptured from the two open sets which are the complements of the two end points. a triangle is captured by the open cover via three open sets which are the complements of the three sides. i.e. each such open set represents the one vertex it contains, and the intersection of two of these represents the one (open) edge it contains, and the intersection of all three represents the face of the triangle.
cech cohomology is defined by assigning coefficients to each such cech polyhedral face, and making some algebraic constructions.
cech sheaf cohomology is defined by using coefficients which assign to each cech "face" the family of functions defined on that open set and having some property, like holomorphicity, or rational functions, etc...
one can also use more general "functions" as coefficients such as sections of a line bundle or vector bundle. In all cases the lebesgue dimension of the space gives the dimension above which the cohomology groups can not be non zero.
Another definition of dimension is based on filtrations by subspaces. in a vector space the dimension is the length of a maximal nested family of subspaces. I.e. point, line containing the point, plane containing the line, etc,... If one gets n+1 such ensted subspaces at most, then the dimension is n.
This generalizes in algebraioc geometry to the definition of the dimension of a noetherian space. I.e. the dimension of an algebraic variety is the length of a maximal nested chain of algebraic subvarieties. pont, curve containing the point, surface containiong the curve, etc etc...
I think a noetherian space is one in which descending chains of closed sets are always finite. oh yesy that's right because in affine algebraic geometry that correspond to ascending chains of ideals in the coordinate ring being finite, the usual meaning of noetherian.
thus the corresonding definition of dimension in a ring, is the maximal length of a nested chain of prime ideals, since these correspond to irreducible sets. oops that emans i forgot to say irreducible in the definition above.
for rings this is called the "krull" dimension.
anyway dimension is a big subject, and a central one in all geometry.
