Justify the examples (e) and (f) in details 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. (e) Let X = [0, 1] U [2, 3] with the subspace topology of the real line.Then A = [0, 1] and B = [2, 3] are disjoint, non-empty open subsetsof X for which X = A U B, so X is disconnected. (Note that A and Bare relatively open since A = (-∞, 3/2) n X and B = (3/2, ∞0) n X.)(f) Let X denote the set of real nụmbers with the one additional pointa = (0, 1), and let the topology T for X consist of Ø and all subsetsof X which contain a. Then there do not exist two disjoint non-emptyopen sets, so X is connected. Note that as a subspace of (X, T ), R isassigned the discrete topology and is therefore disconnected. Thisexample (as well as examples (c), (d), and (e) above) demonstrate thatthe property of being connected is definitely not hereditary.%3D

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Description: Justify the examples (e) and (f) in details 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 pro

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(e) Let X = [0, 1] U [2, 3] with the subspace topology of the real line. Then A = [0, 1] and B = [2, 3] are disjoint, non-empty open subsets of X for which X = A U B, so X is disconnected. (Note that A and B are relatively open since A = (-∞, 3/2) N X and B = (3/2, ∞0) n X.) (f) Let X denote the set of real nụmbers with the one additional point a = (0, 1), and let the topology T for X consist of Ø and all subsets of X which contain a. Then there do not exist two disjoint non-empty open sets, so X is connected. Note that as a subspace of (X, T ), R is assigned the discrete topology and is therefore disconnected. This example (as well as examples (c), (d), and (e) above) demonstrate that the property of being connected is definitely not hereditary. %3D

(e) Let X = [0, 1] U [2, 3] with the subspace topology of the real line. Then A = [0, 1] and B = [2, 3] are disjoint, non-empty open subsets of X for which X = A U B, so X is disconnected. (Note that A and B are relatively open since A = (-∞, 3/2) N X and B = (3/2, ∞0) n X.) (f) Let X denote the set of real nụmbers with the one additional point a = (0, 1), and let the topology T for X consist of Ø and all subsets of X which contain a. Then there do not exist two disjoint non-empty open sets, so X is connected. Note that as a subspace of (X, T ), R is assigned the discrete topology and is therefore disconnected. This example (as well as examples (c), (d), and (e) above) demonstrate that the property of being connected is definitely not hereditary. %3D

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Justify the examples (e) and (f) in details

Definition: A topological space X is disconnected or separated if it is the union
of two disjoint, non-empty open sets. Such a pair A, B of subsets of X is called a
separation of X. A space X is connected provided that it is not disconnected. In other
words, X is connected if there do not exist open subsets A and B of X such that
A + Ø, B+ Ø, ANB= Ø, AUB= X.
%3D
A subspace Y of X is connected provided that it is a connected space when
assigned the subspace topology. The terms connected set and connected subset are
sometimes used to mean connected space and connected subspace, respectively.

(e) Let X = [0, 1] U [2, 3] with the subspace topology of the real line.
Then A = [0, 1] and B = [2, 3] are disjoint, non-empty open subsets
of X for which X = A U B, so X is disconnected. (Note that A and B
are relatively open since A = (-∞, 3/2) n X and B = (3/2, ∞0) n X.)
(f) Let X denote the set of real nụmbers with the one additional point
a = (0, 1), and let the topology T for X consist of Ø and all subsets
of X which contain a. Then there do not exist two disjoint non-empty
open sets, so X is connected. Note that as a subspace of (X, T ), R is
assigned the discrete topology and is therefore disconnected. This
example (as well as examples (c), (d), and (e) above) demonstrate that
the property of being connected is definitely not hereditary.
%3D

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