4-color map proof

[Steven G]

I am aware that two former students of mathematics from Queens College/CUNY went on and proved the 4-color conjecture.
I understand that they concluded that there was a finite number of different maps and then had a computer show that each of these maps could in fact be colored with 4 or less colors.

My question is how can there be a finite number of maps? What if I add one more region to one of these finite number of maps?
I see two cases then:
This extra region is actually a new map or the map with this extra region has already been counted. Now let's keep adding one more region. I feel that one of the cases must always be true. Well if we keep adding a new region one by one an infinite number of times I claim that there will be an infinite number of different maps.
Obviously I am wrong and am hoping that someone can explain what is going on here.

 

[JeffM]

Jomo

It is not the number of areas into which a plane can be divided. It is about the number of ways, if any, five areas in a plane can be mutually adjacent. So the "map" never need consist of more than five areas.

I suspect there are only a limited number of topological ways four closed areas in plane can be mutually adjacent. Then if you can show that no fifth area adjacent to all four is possible in each of those cases, you are done.