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3) Assume that X is a non-empty Baire space, which is Hausdorff. If D C X is a countable dense Gs subset of X, show that there exists x D, which is isolated in X.

3) Assume that X is a non-empty Baire space, which is Hausdorff. If D C X is a countable dense Gs subset of X, show that there exists x D, which is isolated in X.

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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Exercise 3 Part 3!
Exercise 3 1) Assume that X is a Hausdorff space. If D C X is a dense subset of X. If the point xD is an isolated point in D for the induced topology on the subset DC X, show that x is also an isolated point in X. 2) Does the result above remain true if X is not Hausdorff? 3) Assume that X is a non-empty Baire space, which is Hausdorff. If D C X is a countable dense Gs subset of X, show that there exists x D, which is isolated in X.
Transcribed Image Text:Exercise 3 1) Assume that X is a Hausdorff space. If D C X is a dense subset of X. If the point xD is an isolated point in D for the induced topology on the subset DC X, show that x is also an isolated point in X. 2) Does the result above remain true if X is not Hausdorff? 3) Assume that X is a non-empty Baire space, which is Hausdorff. If D C X is a countable dense Gs subset of X, show that there exists x D, which is isolated in X.
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