17. Closed Sets and Limit PointsDefn: The set A is closed iff X − A is open.Thm 17.1: X be a topological space if and onlyif the following conditions hold:(1) ∅, X are closed.(2) Arbitrary intersections of closed sets are closed.(3) Finite unions of closed sets are closed.Note arbitrary intersections of open sets need notbe open. Example: ∩∞n=1(−1n,1n) =Note arbitrary unions of closed sets need not beclosed. Example: ∪∞n=1[1n, 1 −1n] =Thm 17.2: Let Y be a subspace of X. Then aset A is closed in Y if and only if it equals theintersection of a closed set of X with Y .Thm 17.3: Let Y be a subspace of X. If A isclosed in Y and Y is closed in X, then A is closedin X.11Def: The interior of A = Int A = A0= ∪Uopen⊂AUDef: The closure of A = Cl A = A = ∩A⊂FclosedFNote: A is the smallest closed set containing A.Thm 17.4: Let Y be a subspace of X, A ⊂ Y .Let A denote the closure of A in X. Then theclosure of A in Y equals A ∩ Y .Defn: A intersects B if A ∩ B 6= ∅Thm 17.5: Let A be a subset of the topologicalspace X.(a) x ∈ A if and only if (x ∈ UopenimpliesU ∩ A 6= ∅).(b) x ∈ A if and only if (x ∈ B where B is a basiselement implies B ∩ A 6= ∅).Defn: U is a neighborhood of x if U is an openset containing
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