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.
