1. A point a of a topological space A is isolated if the singleton set {a}...

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1. A point a of a topological space A is isolated if the singleton set {a} is open in A. Let I(A) be the set of all isolated points of A. a) When X is a topological space and A is a subspace of X show that the closure of A is the disjoint union of I(A) and the limit set of A, that is show that A'U I(A) and show that A'I(A) is empty. b) If A is a subspace of Euclidean n-space show that I(A) is countable. Hint: construct disjoint balls in n-space so that each meets I(A) at just one point. 1. A point a of a topological space A is isolated if the singleton set {a} is open in A. Let I(A) be the set of all isolated points of A. a) When X is a topological space and A is a subspace of X show that the closure of A is the disjoint union of I(A) and the limit set of A, that is show that A'U I(A) and show that A'I(A) is empty. b) If A is a subspace of Euclidean n-space show that I(A) is countable. Hint: construct disjoint balls in n-space so that each meets I(A) at just one point.

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a To show that the closure of A is the disjoint union of IA and the limit set of A we first note tha... View the full answer