# compactness theorem

• May 26th 2010, 02:10 AM
nngktr
compactness theorem
The Four Color Theorem asserts that if a region in the plane is divided into ﬁnitely many countries, then each country may be colored either
red, green, blue, or yellow in such a way that no two countries with a common border (of positive length) get the same color. Use the Compactness
Theorem to show that this remains true even if there are inﬁnitely many
countries.

I get the general idea of this problem. But I'm not sure how to quantify the four color theorem properly. Could anyone please help?
• May 27th 2010, 10:46 AM
kompik
Quote:

Originally Posted by nngktr
The Four Color Theorem asserts that if a region in the plane is divided into ﬁnitely many countries, then each country may be colored either
red, green, blue, or yellow in such a way that no two countries with a common border (of positive length) get the same color. Use the Compactness
Theorem to show that this remains true even if there are inﬁnitely many
countries.

I get the general idea of this problem. But I'm not sure how to quantify the four color theorem properly. Could anyone please help?

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