Prove the following Proposition: Let X be a topological space and let A ⊆ X be any subset. The following are equivalent: • A is closed in X.• A = closure(A) in X• Every point of X\A has a neighborhood contained in X\A. • A contains all of its limit points.

Prove the following Proposition: Let X be a topological space and let A ⊆ X be any subset. The following are equivalent: • A is closed in X.• A = closure(A) in X• Every point of X\A has a neighborhood contained in X\A. • A contains all of its limit points.

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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Question

Prove the following Proposition:

Let X be a topological space and let A ⊆ X be any subset. The following are equivalent:

• A is closed in X.
• A = closure(A) in X
• Every point of X\A has a neighborhood contained in X\A.

• A contains all of its limit points.

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