By George Gabriel Stokes

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16. Let X be a topological space, and let A ⊆ X. The closure of A, written A− , is the set A ∪ Ad . It is clear that the intersection of the members of the family of all closed sets containing A is the closure of A. 15. Let X be a topological space, and let A ⊆ X, then A−− = A− . Proof. 16 A−− = A− ∪ (A− )d . We show that (A− )d ⊆ A− . 3 there exists some U ∈ Ux , such that U ∩ A = ∅. Select O, open, such that x ∈ O ⊆ U, then O ∈ Ux , and further since O ∩ A ⊆ U ∩ A = ∅, we have O ∩ A = ∅. Now since x ∈ (A− )d , O ∩ A− contains some point y = x.

Some examples of Hasse diagrams of partially ordered sets are shown below. The third example, represent a totally ordered set. We now define quasi-ordered relation which is weaker than partially ordered relation. 7. A quasi-ordered set (S, ≤) is a set S together with a binary relation R≤ on S satisfying the following conditions, where for a, b ∈ S, a ≤ b simply denotes (a, b) ∈ R≤ : (1) a ≤ a; (2) If a ≤ b and b ≤ c, then a ≤ c. (reflexivity) (transitivity) “≤” is called a quasi-ordered relation on set S.

4 Countable Sets A set is finite if and only it can be put into one-to-one map with a set of the form {p|p ∈ N and p < q} for some q ∈ N, where N is the set of non-negative integers. 4. A set A is called countably infinite if and only if it can be put into one-to-one map with the set N of non-negative integers; A set is countable if and only if it is either finite or countably infinite, otherwise it is called uncountable . 3. A subset of a countable set is countable. Its proof remains as an exercise.