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Theorem 3.60. Let X be a Hausdorff space. If x and y are sequential limit points of X with respect to the same sequence, then x = y.

Theorem 3.60. Let X be a Hausdorff space. If x and y are sequential limit points of X with respect to the same sequence, then x = y.

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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Solve Theorem 3.60 in detail please 

i posted this question on chegg and received the wrong answer so please only attempt the question when you know answer.

Theorem 3.60. Let X be a Hausdorff space. If x and y are sequential limit points of X with respect to the same sequence, then x = y. Problem 3.61. Show (by example) that the assumption that X is a Hausdorff space in Theorem 3.60 is necessary. Problem 3.62. Is there an example of a Hausdorff space X, a subset Y C X, and a point x € X such that x is a limit point of Y, but not a sequential limit point of Y ?
Transcribed Image Text:Theorem 3.60. Let X be a Hausdorff space. If x and y are sequential limit points of X with respect to the same sequence, then x = y. Problem 3.61. Show (by example) that the assumption that X is a Hausdorff space in Theorem 3.60 is necessary. Problem 3.62. Is there an example of a Hausdorff space X, a subset Y C X, and a point x € X such that x is a limit point of Y, but not a sequential limit point of Y ?
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