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Compactness and Locally Compact Spaces in Hausdorff Spaces

The lecture covers various theorems related to compactness in Hausdorff spaces, including the separation of points by open sets, the compactness and closedness of subspaces, and the properties of homeomorphisms. It introduces the concept of locally compact spaces at specific points, discussing how every compact space is locally compact but not vice versa. An example with the real line illustrates local compactness, and a proof is provided to establish that every compact space is locally compact. Additionally, it explores the relationship between open continuous surjections and the local compactness of spaces.

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Compactness and Locally Compact Spaces in Hausdorff Spaces

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  1. Set Topology MTH 251 Lecture # 30

  2. Lecture Plan Compactness in Hausdorff spaces Locally compact spaces • •

  3. Theorem. If X is a Hausdorff space and C, a compact subspace of X such that ? ∉ ? then ? and ? can be separated by open sets.

  4. Theorem. Theorem. Every compact subspace of a Hausdorff space is closed.

  5. Theorem. Let ?:? → ? be a bijective continuous function . If ? is compact and ? is Hausdorff, then ? is homeomorphism. • Discussion.

  6. Definition 1 (Locally Compact) Definition 1 (Locally Compact) A space X is said to be locally compact at x ∈ X •

  7. Definition 1 (Locally Compact) Definition 1 (Locally Compact) A space X is said to be locally compact at x ∈ X • If there is some compact subspace Y of X that contains a nbd of • x.

  8. Definition 1 (Locally Compact) Definition 1 (Locally Compact) A space X is said to be locally compact at x ∈ X • If there is some compact subspace Y of X that contains a nbd of • x. If X is locally compact at each of its points, then X is called a • locally compact space.

  9. Renark. Every compact space is locally compact but converse is not true in general.

  10. Renark. Every compact space is locally compact but converse is not true in general. • Example.

  11. Renark. Every compact space is locally compact but converse is not true in general. • Example. The real line ? is locally compact. •

  12. Renark. Every compact space is locally compact but converse is not true in general. • Example. The real line ? is locally compact. • The point ? lies in some interval (? − ?, ? + ?), which in tern is • contained in the compact subspace {? − ?, ? + ?}

  13. Theorem. Prove that every compact space is locally compact.

  14. Theorem. Let ?:? → ? be an open continuous surjection. If ? is locally compact then prove that ? is locally compact.

  15. Good Luck

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