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Unit 6: Closed Sets and Limit Point

6.4 Keywords Notes

Discrete Topology: Let X be any non-empty set and T be the collection of all subsets of X. Then T
is called discrete topology on the set X.

Open and Closed Set: Let (X, T) be a topological space. Any set A  T is called an open set and
X  A is a closed set.
Subspace: Let (X, T) be a topological space and a subset S of X, the subspace topology on S is

defined by T  S  U|U T .
S
6.5 Review Questions

1. Let X be a topological space and A be a subset of X. Then prove that A is the smallest
closed set containing A.

2. Prove that A is closed iff A  A.
3. Let (Y, U)  (X, S) and A  Y. Prove that A point y  Y is U-limit point of A iff y is a T-limit
point of A.

4. Show that every closed set in a topological space is the disjoint union of its set of isolated
points and its set of limit points, in the sense that it contains these sets.
5. Show that if U is open in X and A is closed in X, then U  A is open in X, and A  U is closed
in X.

6.6 Further Readings

Books J. L. Kelley, General Topology, Van Nostrand, Reinhold Co., New York.
S. Willard, General Topology, Addison-Wesley, Mass. 1970.

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