Justify the examples (c) and (d) in details (c) Let Ybe the set of non-zero real numbers with the subspace topologyof R. Then Y is disconnected since (-∞, 0) and (0, ∞) form a sep-aration.(d) Let Z = R²\R denote the plane minus the real axis, with the subspacetopology. Then Z is disconnected since the upper half plane U{(x,, x2) E R²: x2 > 0} and lower half plane V = {(x1, x2) E R²:X2 < 0} form a separation.%3D Definition: A topological space X is disconnected or separated if it is the unionof two disjoint, non-empty open sets. Such a pair A, B of subsets of X is called aseparation of X. A space X is connected provided that it is not disconnected. In otherwords, X is connected if there do not exist open subsets A and B of X such thatA + Ø, B+ Ø, ANB= Ø, AUB= X.%3DA subspace Y of X is connected provided that it is a connected space whenassigned the subspace topology. The terms connected set and connected subset aresometimes used to mean connected space and connected subspace, respectively.

Name: Justify the examples (c) and (d) in details (c) Let Ybe the set of non
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Description: Justify the examples (c) and (d) in details (c) Let Ybe the set of non-zero real numbers with the subspace topologyof R. Then Y is disconnected since (-∞, 0) and (0, ∞) form a sep-aration.(d) Let Z = R²\R denote the plane minus the real axis, with th

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(c) Let Y be the set of non-zero real numbers with the subspace topology of R. Then Y is disconnected since (–∞, 0) and (0, ∞0) form a sep- aration. (d) Let Z = R²\R denote the plane minus the real axis, with the subspace topology. Then Z is disconnected since the upper half plane U {(x1, x2) E R?: x, > 0} and lower half plane V = {(x,, x2) E R²: X2 < 0} form a separation. %3D