Can the borders of a map be deformed to give arbitrary area to any region? Let's say I have a geographic map, a connected region divided into sub-regions. Is it possible to deform the map (the borders of the regions) so that each sub-region is of arbitrary area while maintaining the adjacencies?
I think it is possible, but I've forgotten almost all my topology. Is this a theorem? (Or basic definition?) Also, what is the correct way to describe how the original map and its deformation are related, does "homeomorphic" apply here?

By intuition:


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*Imagine your map to have all regions with borders made of very elastic rubber 

*Shrink down your entire map to have all regions areas smaller than the smallest area you want

*Start with a region and "blow" air into it to make it as big as you need. After that, make the region's borders rigid. This way if you deform another region of the map, the rigid region will maintain its area

*One by one, continue with the other regions and do the same: expand and make it rigid


Links that may help:


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*http://en.wikipedia.org/wiki/F%C3%A1ry's_theorem

*enter link description here
