Defining topological spaces with the notion of continuous path

Question

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.

Answer 1

I don't think I've seen a definition of a space-like notion phrased only in terms of paths, but you could certainly write one down, perhaps as an example of concrete sheaves. It seems related to the notion of Froelicher space which defines "smoothness" in terms of paths and "co-paths."

However, it seems unlikely to me that you'll get a very good notion without including also some higher-dimensional test objects in addition to intervals. For instance, there is the notion of $\Delta$-generated space, which can be described as a set $X$ with sets of distinguished maps $\Delta^n \to X$ for all $n$, satisfying a bunch of properties. I don't know any description of those properties other than "if $X$ is given the final topology induced by the distingished maps, then all continuous maps $\Delta^n \to X$ are distinguished", but that doesn't mean there isn't one; I don't know whether anyone has looked.

Answer 2

Try the wikipedia entry on sequential spaces. I quote:

"Many conditions have been shown to be equivalent to $X$ being sequential. Here are a few:

$X$ is the quotient of a first countable space.

$X$ is the quotient of a metric space.

For every topological space $Y$ and every map $f : X \to Y$, we have that f is continuous if and only if for every sequence of points (xn) in X converging to x, we have (f(xn)) converging to f(x).

The final equivalent condition shows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences in the space."

The point is that given a set of test maps from $I$ to a set $X$ you can put a topology on $X$ as the quotient of the disjoint union of copies of $I$ one for each test map. And then $X$ will be the quotient of a first countable space.

Answer 3

No such definition can be given for a general topological space, because a general topological space can have very few morphisms from or into $[0;1]$. E.g. consider $\Bbb Q$: there are no non-trivial continuous paths, containing only rational points. On the other side, Urysohn's lemma states that only normal topological spaces have enough continuous functions on them.

Your proposed axiomatization looks very similar to the axioms of $(\infty,1)$-categories. There you have objects, corresponding to points, morphisms between them, corresponding to paths, 2-morphisms between morphisms, corresponding to homotopies between paths etc. Every topological space has a canonically associated $(\infty,1)$-category, constructed just as above. There is an adjunction between the bicategories of topological spaces and of $(\infty,1)$-categories. In fact, it is even a Quillen equivalence (equivalence of homotopy categories) for some natural choices of model structure on categories (i.e. for some notion of homotopy equivalence). However, the image of this adjunction in $\mathcal{T}op$ consists not of all spaces, but only of CW-complexes. So the answer is: it works, but only for very nice spaces.

You can look further information on $(\infty,1)$-categories and quasicategories. A short survey of theory can be found in the first chapter of J. Lurie's "Higher topos theory".