1. ## 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?

2. 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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