Let’s consider the following pseudo-definition of (nice) topological spaces : a space is a set $X$ together with distinguished paths $[0,1]\to{}X$ called continuous paths, distinguished maps $[0,1]\times[0,1]\to{}X$ called homotopies, and so on in every dimension (in a globular style), satisfying a bunch of properties. For example, the constant path should be continuous, the composition of two continuous paths is continuous, the slices of an homotopy are continuous paths, etc.

**Is there a way to formalize precisely this definition, and how?**

I apologize for the vagueness of the question.

And this is just a curiosity, I had to talk about topology to computer scientists a few days ago and the most intuitive definition of topological space I found is to say that a topological space is a set with a well-behaved notion of continuous path. And now I’m wondering whether this could be an honest definition of topological spaces.

joiningof two continuous paths is continuous"? $\hspace{2 in}$ Composition of paths would in general not make sense. $\;\;$ $\endgroup$ – user5810 Jan 27 '12 at 4:13