= Solution
At every regular point of the <Riemann sphere>, there is a neighborhood containing no <poles>; at each <pole>, there is a neighborhood containing no other <poles>. These neighborhoods cover the sphere. Its <compactness> gives a finite subcover, each member containing at most one <pole>. Hence \b[the total number of <poles>, including a possible <pole> at infinity, is finite]. Equivalently, an infinite <pole> set would have an accumulation point on the compact sphere, contrary to local regularity or isolation there.
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