What makes Euclidian space univalent and topological space multivalent? Here is a quote from wikipedia:

Euclidean axioms leave no freedom, they determine uniquely all
  geometric properties of the space. More exactly: all three-dimensional
  Euclidean spaces are mutually isomorphic. In this sense we have "the"
  three-dimensional Euclidean space. In terms of Bourbaki, the
  corresponding theory is univalent. In contrast, topological spaces are
  generally non-isomorphic, their theory is multivalent. A similar idea
  occurs in mathematical logic: a theory is called categorical if all
  its models of the same cardinality are mutually isomorphic. According
  to Bourbaki,[5] the study of multivalent theories is the most striking
  feature which distinguishes modern mathematics from classical
  mathematics

What is the underlying argument here?
What makes it so that we can speak of the Euclidian space?
 A: When using axiom to specify what one wants to talk about, there are generally two possible intentions behind this:


*

*One can use axioms to try specifying one and only one structure. One tries to find the axioms which describe the wanted object uniquely, pin it down to study the properties of it and nothing else. This, for example, was the intention for construction the Euclidean space (e.g. using Tarski's axioms). The axioms were chosen in a way so that any other structure satisfying them too is 
essentially the same thing (i.e. isomorphic). In the case of Euclidean space this was successful. This is why we can speak of the Euclidean space. But it does not work for everything we can think about. According to Gödel's incompleteness theorem it is not possible to pin down the natural numbers uniquely. There will always be non-isomorphic objects that satisfy Peanos's axioms equally well. Even tho the intention is the same here, it is damned to fail. This however does not prevent most of the mathematicians to talk about the natural numbers. It seems the intention is leading here.

*On the other hand, axioms can be used to only vaguely describe what one wants to talk about. Maybe we want to study a whole class of objects at the same time. For example, if I do not only want to study the Euclidean space, but want to find any theorem that is true in any Euclidean geometry in general, i.e. in any dimensions. Those axioms must leave room for several objects, here for general $n$-dimensional spaces for any $n\in \Bbb N$. And this happends all the time. Mathematicians love to study several objects at the same time, because they love the general applicability of their tools. The axioms of group theory are so vague, that they can describe all kind of number spaces, vector spaces, finite and infinite objects, Rubic's cubes, particle physics, etc. There is not the topological space because it was the intention to let room for all kinds of spaces that behave in a specific similar way. Take any Euclidean geometry, it is a topological space. Any theorem proven for topological spaces must hold in any dimension. But also spheres, tori, lines, dots, actually any set can be used to build a topological space and can be studied by using the same few axioms.
In short, the resason why we can say the Euclidean space but not the topological space is because the former system (theory) was designed to only fit to a single object (and it was proven that this is the case) while the latter was designed to fit a whole class of structures. It is all about the intention. 
And I believe the shift in modern mathematics to the second way is because of the discovery of the power of generalization. As made clear above, in this way it is possible to describe myriads of seemingly different objects inside a single theory. We can prove things for the natural numbers and the Euclidean space in a single theorem, while in the past these two fields are mostly distinct and have to be studied seperately.
