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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.
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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