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 topological dimension, 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
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