Let (X, d) be a metric space. A subset E of X is called pathwise connected if for every e₁, e2 € E, there exists a continuous function e: [0, 1] → E such that e(0) = e₁ and e(1) = e2. Prove that if E C X is pathwise connected then E is connected.
Let (X, d) be a metric space. A subset E of X is called pathwise connected if for every e₁, e2 € E, there exists a continuous function e: [0, 1] → E such that e(0) = e₁ and e(1) = e2. Prove that if E C X is pathwise connected then E is connected.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![Let (X, d) be a metric space. A subset E of X is called pathwise connected if
for every €₁, €2 € E, there exists a continuous function e : [0, 1] → E such that
e (0) = e₁ and e(1) = €2. Prove that if E C X is pathwise connected then E is
connected.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F9e8f8bfe-8aa2-4298-b383-987eee00e7ab%2Fvwgmobl_processed.png&w=3840&q=75)
Transcribed Image Text:Let (X, d) be a metric space. A subset E of X is called pathwise connected if
for every €₁, €2 € E, there exists a continuous function e : [0, 1] → E such that
e (0) = e₁ and e(1) = €2. Prove that if E C X is pathwise connected then E is
connected.
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