Basic Topology Please check my solution to Q7, I was told there is a counter example to that solution?? Kindly show how and what is the correct solution SetDerived of every# Lef A'So 7SefContainthisthe setCimitofpoints ofwe have to show that A' is closed.XE (A¹) C beLetanisaclosed set(Q denote the set ofnaSet A.abitrany pointneighbourhood of X, NEGx) Sot NE(x) notlimit pant of ANE (x) CACmeansSince every point of NE (x) is an interior point of NeC)NE (x) not contain any point of ASO, NE (x) ≤ (A¹) Cthen (A)C is open23So A is closed.9. Consider the set of real number IR with Standardon Ron R. (usual topologyfopologyLefA = Qrational number) For each of the following, if the statement about a topological space is alwaystrue, prove it; otherwise, give a counterexample.7. The derived set of every set is a closed set.8. The boundary of every set is a closed set.9. The boundary of every set has empty interior.10. Every nonempty closed set with empty interior is the boundary of some set.11. If a set has empty interior, then so does its closure.12. The closure of a set coincides with the closure of its interior. "13. The interior of a set coincides with the interior of its closure..14. Every set that does not meet its boundary is open.2 Base for a Topologynext introduce the idea of a base for a topology, a concept that yields aain economy of thought and effort in defining a topology for a set.DefinitionLot (YT) be a tonological space. A subset B of T such that every element

Answered: Image /qna-images/answer/b2ed72f6-4370-4cf1-b064-16ea823f40de.jpg

Set Berived of every set is a closed set Lef A' is the set Cimit of we have to show that A' is closed. . Let XE € (A¹) ² be an abitrany point neighbourhood So 7 Contain limit pänt of A this means points of a Set A of X, NEGx) Sit NEC NE (₂) CAC Since every point of NE (x) is an interior point NEGx) not Contain any point of A C Calc

Set Berived of every set is a closed set Lef A' is the set Cimit of we have to show that A' is closed. . Let XE € (A¹) ² be an abitrany point neighbourhood So 7 Contain limit pänt of A this means points of a Set A of X, NEGx) Sit NEC NE (₂) CAC Since every point of NE (x) is an interior point NEGx) not Contain any point of A C Calc

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.1: Sets And Geometry

Problem 11E: For the sets gives in Exercise 9, is there a distributive relationship for union with respect with...

See similar textbooks