Maybe my first attempt to ask the following question was a bit confusing, so let me try it again, without ado:
(How) can – in a 2-dimensional topological space – the concept of two curves crossing (instead of just touching) at a point $x$ be defined in purely topological terms?
My own handwaving attempt: In appropriate path-connected 2-dimensional spaces there are loops small enough to split the space in two. (Right?) If one cuts out a small enough piece around $x$ of one of the two curves one can join its two ends resulting in such a small enough loop. (Right?) This loop locally defines an "inside" and an "outside". If in every neighbourhood of $x$ the other curve has points in both of these regions, it crosses the first curve. Otherwise it only touches it.
(If the above attempt is sound: do you consider it elementary/basic/trivial? At least it seems to involve Jordan's curve theorem.)