How do I show (4),(5),(6) in 8.1? Definition. Let X be a topological space. Then X is connected if and only if X is notthe union of two disjoint non-empty open sets.Definition. Let X be a topological space. Subsets A, B in X are separated if and onlyif AnB = An B = Ø. Thus B does not contain any limit points of A, and A does notcontain anylimit points of B. The notation X =A | B means X = A U B and A and Bare separated sets.Theorem 8.1. The following are equivalent:(1) X is connected.(2) There is no continuous function f : X → Rstd such that f(X) = {0,1}.(3) X is not the union of two disjoint non-empty separated sets.(4) X is not the union of two disjoint non-empty closed sets.(5) The only subsets of X that are both closed and open in X are the empty set and X itself.(6) For every pair of points p and q and every open cover {U«}aea 0f X there exist a finitenumber of the Uga's, {U«,, Uaz, Uaz,.., Uan} such that pE Ua, q E Uan' and for eachi < n, Ua, n Ua41 + Ø.Ci+1

as given all the conditions are equivalent (4) To show - X is not the union of two disjoint non…

(3) X is not the union of two disjoint non-empty separated sets. (4) X is not the union of two disjoint non-empty closed sets. (5) The only subsets of X that are both closed and open in X are the empty set and X itself. (6) For every pair of points p and q and every open cover {Uq}aea of X there exist a finite number of the Uq's, {U«,, Uaz, Uaz»….. , Uan} such that p E Ua,, q E Ugn; and for each i < n, Ua; n Uai+1 + Ø.

(3) X is not the union of two disjoint non-empty separated sets. (4) X is not the union of two disjoint non-empty closed sets. (5) The only subsets of X that are both closed and open in X are the empty set and X itself. (6) For every pair of points p and q and every open cover {Uq}aea of X there exist a finite number of the Uq's, {U«,, Uaz, Uaz»….. , Uan} such that p E Ua,, q E Ugn; and for each i < n, Ua; n Uai+1 + Ø.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 13E: 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is...

See similar textbooks