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5 Example. Let (X, τ ) be an indiscrete space and A a subset of X with at least two elements. Then it is readily seen that every point of X is a limit point of A. ) The next proposition provides a useful way of testing whether a set is closed or not. 6 Proposition. Let A be a subset of a topological space (X, τ ). Then A is closed in (X, τ ) if and only if A contains all of its limit points. Proof. We are required to prove that A is closed in (X, τ ) if and only if A contains all of its limit points; that is, we have to show that (i) if A is a closed set, then it contains all of its limit points, and (ii) if A contains all of its limit points, then it is a closed set.

Y .......... .......... 4 are satisfied. Thus B is indeed a basis for the euclidean topology on R2 . 9 we defined a basis for the euclidean topology to be the collection of all “open rectangles” (with sides parallel to the axes). 5 shows that “open rectangles” can be replaced by “open equilateral triangles” (with base parallel to the X-axis) without changing the topology.

X ∈ X is said to be a limit point (or accumulation point or cluster point) of A if every open set, U , containing x contains a point of A different from x. 2 Example. Consider the topological space (X, τ ) where the set X = {a, b, c, d, e}, the topology τ = {X, Ø, {a}, {c, d}, {a, c, d}, {b, c, d, e}}, and A = {a, b, c}. Then b, d, and e are limit points of A but a and c are not limit points of A. Proof. The point a is a limit point of A if and only if every open set containing a contains another point of the set A.

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