Density of sets
◆A POINT C IS SAID TO BE A CLUSTER POINT OF A SUBSET S OF REAL NUMBERS, IF TAKING ANY SET N=( C-∆ ,C+∆ ) SURROUNDING C , N CONTAINS INFINITELY MANY POINTS OF THE SET S , WHERE ∆>0 .
◆AND , A POINT I IS SAID TO BE AN INTERIOR POINT OF A SET T OF REAL NUMBERS, IF THERE EXISTS SOME ∆>0 SUCH THAT (I-∆, I+∆) IS A SUBSET OF T .
◆A SUBSET U OF REAL NUMBERS (R) IS SAID TO BE CLOSED IN R IF ALL CLUSTER POINTS OF U REMAIN IN U .
◆AND , A SUBSET V OF R IS SAID TO BE OPEN IN R IF ALL INTERIOR POINTS OF V REMAIN IN V .
NOW LET D= { p^m.q^n : m, n are integers and p,q are positive integers with (p, q) = 1 } AND G= -D , IS :
THEN SET OF ALL CLUSTER POINTS OF THE SET D U G IS :
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1 solution
Note that , in the set D= {p^m.q^n : m,n are integers and p, q are co-prime positive integers} , then for any p^m.q^n in D, p^ m.q^n = exp(m.lnp + n.lnq) =exp(lnp(m+n.lnq/lnp)) = p^ (m + lnq/lnp) , now as (p,q) =1, hence lnq/lnp is irrational , let i= lnq/lnp ; hence p^m.q^n = p^ (m+ni) , then for any r>0 , r = p^ ( logr / logp ) =p^x , now by DIRICHLET'S IRRATIONALITY CRITERION , x is a cluster point of some sequence of the set {m+ni : i is irrational and m, n are integers } , but as the function f(x) = p^x , p>0 , is continuous in everywhere, then there exists some sequence in D converging to r , but as r is arbitrary positive real , hence D is dense in the set of all positive reals , similarly G is dense in set of all negative reals , hence D U G is dense in the set of all reals , so each real number is a cluster point of the set D U G , but the set of all reals is both open and closed in R .