What is the exact definition of the dimension of a topological space?
?? Your title is "the definition of dimension" but your question is "what is the exact definition of a topological space?" Which is it? The definition of "dimension" depends strongly on exactly what kind of space you are dealing with. The definition of "topological space", however, is quite simple:
A topological space is a set with a topology!
And a topology (for set X) is a collection, T, of subsets of X satisfying:
The empty set is in T.
The entire set X is in T.
The union of any collection of sets in T is also in T.
The intersection of any finite collection of sets in T is also in T.
For any set X, whatsoever, the following are topologies on T:
The collection of all subsets of X. (Often called the "discrete" topology.)
The collection containing only the empty set and X. (Often called the "indiscrete" topology.)
I made a mistake. What I want to ask is "the definition of the dimension of a topological space"
There are many notions of dimension, as HallsofIvy warned you. I think the most general one is the http://en.wikipedia.org/wiki/Lebesgue_covering_dimension" [Broken], also known as the covering dimension.
The "Hausdorff dimension" and "Box counting dimension"
are other "dimensions" that can be used.
Those ones require a metric though. A non-metrizable space should still have a topological dimension.
Other useful dimensions are upper and lower Minkowski dimensions (related to the box-counting dimension) and the Assouad dimension (aka Bouligand dimension).
It's interesting to note that the topological dimension of a space is also equal to the infimum the Hausdorff dimensions of all spaces to which it is homeomorphic, in the case when your space is a separable metric space (so that this makes sense). I think it's pretty cool that those two ways of defining topological dimension give the same number! (I have no idea how to prove it. I don't think it's easy.)
cohomological dimension is another cool definition.
Ditto Xevarion; clearly the OP wants the Lebesgue covering dimension. Many good "general topology" textbooks cover this--- er, no pun intended
Separate names with a comma.