True or False If E R and x epsilon E then either x is an acc

True or False? If E R and x epsilon E, then either x is an accumulation point of F or x is an isolated point of E. If TRUE, provide a proof. If FALSE, provide a counterexample. True or False? Can a set E, with at least one interior point, have ONLY countably many interior points (not uncountably-many)? If YES, give a specific example. If NO, provide a short proof.

Solution

1) Ans; (TRUE)

Proof: Given that E is an subset of R. and x belongs to E.

since E is a subset of R .if E is equalt to R than x is an accumulation point of E ;becouse every point of R is an accumulation point..

Now if E is subset of R such that ( N -set of natural number or Z -set of integers ) mens that E= N or E= Z.than the point x of E is an isolated point of E.

2) (False)

proof :if a set E with at lesst one interior point then every neighbourhood of that point contains uncountable number of points of the set .so if x belongs to E and x is an interior poin of E than every nbd of x contains uncountable number of pointsE.