Is the number of Lebesgue covering dimensions a topological property? Is it correct to characterize the number of dimensions that a topological space has as a 'topological property'? By 'dimensions,' I have in mind Lebesgue covering dimensions.
If the answer is yes, then why? If the answer is no, then what kind of a property is the number of dimensions?
Isn’t it obvious from the definition that the covering dimension is a topological invariant? it’s defined purely in terms of open sets and set-theory concepts.
There are many different definitions of dimension in different contexts. Some are topological, most are not.  Look up: topological dimension.