Let X be a cofinite topological space. Then show that the derived set of A(denoted as A') is closed for any subset A of X.

Solution:

To show that the derived set A' is closed for any subset A in a cofinite topological space X, we need to demonstrate that the complement of A', denoted as (A')', is open. In other words, we need to show that for any point x in X that is not in A', there exists an open set containing x that is entirely contained within (A')'.

Let's proceed with the proof:

1)  Let x be a point in X that is not in A'. This means x is either not in A or is not a limit point of A.

2)  If x is not in A, then we can choose an open set U containing x such that U ∩ A = ∅ (the empty set). This is possible because X is a cofinite space, meaning that every proper subset of X is finite. Therefore, the complement of A, denoted as X - A, is finite. We can choose U to be X - A, which is open since it is the complement of a finite set.

3)  Now, suppose x is in A but is not a limit point of A. This means there exists an open set V containing x such that V ∩ (A - {x}) = ∅. In other words, V contains no points of A other than x. This is also possible because X is a cofinite space. Since A is a subset of X, A - {x} is a proper subset of X, and therefore finite. We can choose V to be X - (A - {x}), which is open since it is the complement of a finite set.

4)  In both cases, we have found an open set (either U or V) containing x that is entirely contained within the complement of A'. Therefore, (A')' is open, and by definition, A' is closed.

Hence, we have shown that the derived set A' is closed for any subset A in a cofinite topological space X.