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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.)

Edit: Is this "definition" equivalent to the "official" one which I just found here: transversal intersection? And thus, is "appropriate" - see above - just "being a manifold"?

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"...loops small enough to split the space in two. (Right?)" You are thinking about a class of spaces that is locally contractible or some such (e.g. manifolds). It certainly need not hold for pathological spaces. Altogether, for the purposes of intersections, it seems to me it would be better to specialize this question to the category of manifolds where these notions are traditionally studied. – Marek Mar 14 '13 at 19:08
Regarding the rest of the OP, I think your construction works (of course, assuming we are working in some nice category of spaces and with reasonable notion of curves in those spaces), but am not at all sure what one gains by this viewpoint. In any case, what is your question? Do you just want to know how to rigorously prove this starting from some hypotheses on the space and the curves? – Marek Mar 14 '13 at 19:15

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