2 N-Dimensional Space Intersection

[LLT]

ok...
first of all, I was discussing with my friend, he propose an argument of, if you have two N-Dimensional Spaces, (let's called it Sn1 and Sn2), they will form another M-Dimensional Spaces, which M is either 0 or M bigger or equal one smaller or equal N-1...

what he said was that if Sn1 and Sn2 got 1 common dimension, it'll intersect in a line, if they got 2 common dimentsion, it'll intersect in a plane, if 3, a 3D space etc... etc...

I dun think this works... But how can I disproved it?

(he knows it doesn't work with 2 planes, coz u can't produce a point with 2 planars intersection, hence he's trying to convince me it works for N greater or equal to 3)

 

[Almsco]

That's an interesting question. I think the best way to disprove your friend's argument would be to provide a counter-example where two N-dimensional spaces intersect in something that is not an (M-1) dimensional space. For example, if you have two three-dimensional spaces intersecting in one point, then they would form a 0-dimensional space (the point). This is different from what your friend proposed, since he said that M would either have to be 0 or M bigger or equal one smaller or equal N-1. Hope this helps!