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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 \( D \subset X \) is a dense subset of \( X \). If the point \( x \in D \) is an isolated point in \( D \) for the induced topology on the subset \( D \subset X \), 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 \( D \subset X \) is a countable dense \( G_\delta \) subset of \( X \), show that there exists \( x \in D \), which is isolated in \( X \).
Transcribed Image Text:**Exercise 3** 1) Assume that \( X \) is a Hausdorff space. If \( D \subset X \) is a dense subset of \( X \). If the point \( x \in D \) is an isolated point in \( D \) for the induced topology on the subset \( D \subset X \), 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 \( D \subset X \) is a countable dense \( G_\delta \) subset of \( X \), show that there exists \( x \in D \), which is isolated in \( X \).
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