Suppose (X, d) is a metric space and E a non-empty subset of X. Prove that E is disconnected if and only if we can find open sets G₁ and G₂ that satisfy the following properties. ECG₁UG2; • G₁ NE ‡ Ø and G₂ Ñ E ‡ Ø; • (G₁G₂) NE = 0.
Suppose (X, d) is a metric space and E a non-empty subset of X. Prove that E is disconnected if and only if we can find open sets G₁ and G₂ that satisfy the following properties. ECG₁UG2; • G₁ NE ‡ Ø and G₂ Ñ E ‡ Ø; • (G₁G₂) NE = 0.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter7: Real And Complex Numbers
Section7.2: Complex Numbers And Quaternions
Problem 48E
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Please teach me how to prove this! Please show each step and reason why!
Transcribed Image Text:Suppose (X, d) is a metric space and E a non-empty subset of X. Prove that
E is disconnected if and only if we can find open sets G₁ and G₂ that satisfy
the following properties.
• ECG₁ UG2;
• G₁ NE ‡ Ø) and G₂ ^ E ‡ Ø;
• (G₁ NG₂) NE = Ø.
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