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Understanding Compactness in Topological Spaces

Learn about compactness in topological spaces, including definitions, examples, theorems, and the concept of open covering. Discover how finite and countable subcovers play a role in compactness.

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Understanding Compactness in Topological Spaces

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  1. Set TopologyMTH 251 Lecture # 27

  2. Lecture Plan • Compactness • Definition • Examples • Theorems

  3. Compactness In Topological Spaces Definition ( Open Cover ) • Let X be a topological space

  4. Compactness In Topological Spaces Definition ( Open Cover ) • Let X be a topological space • A class F= {iI} of open subsets of X is called an open convering or cover of X,

  5. Compactness In Topological Spaces Definition ( Open Cover ) • Let X be a topological space. • A class F= {iI} of open subsets of X is called an open convering or cover of X, • If =X, i I

  6. Compactness In Topological Spaces Definition ( Open Cover ) • Let X be a topological space. • A class F= {iI} of open subsets of X is called an open convering or cover of X, • If =X, i I. • A class G= {i}, I, of open subsets of X is called a subcover of the cover F,

  7. Compactness In Topological Spaces Definition ( Open Cover ) • Let X be a topological space. • A class F= {iI} of open subsets of X is called an open convering or cover of X, • If =X, i I. • A class G= {i}, I, of open subsets of X is called a subcover of the cover F, • If =X, i.

  8. Compactness In Topological Spaces Definition ( Open Cover ) • Let X be a topological space. • A class F= {iI} of open subsets of X is called an open convering or cover of X, • If =X, i I. • A class G= {i}, I, of open subsets of X is called a subcover of the cover F, • If =X, i. • It is called a finite (respt. countable ) subcover, if is finite (respt. countable).

  9. Example. Consider the class of is the open disc in the plane with radius 1 and centre, . Then is a cover of , i.e., every point in belongs to at least one member of . • On the other hand , the class of open discs Where is not a cover of . () but not belong to any member of H.

  10. Definition (Compactness) • A space X is said to be compact, • if every open cover of X has a finite subcover.

  11. Example. The interval on the real line with the usual topology is not compact.

  12. Definition (Compact subset) • A space X is said to be compact,

  13. Definition (Compact subset) • A space X is said to be compact, • if every open cover of X has a finite subcover.

  14. Definition (Compact subset) • A space X is said to be compact, • if every open cover of X has a finite subcover. • A subset A of a space X is compact,

  15. Definition (Compact subset) • A space X is said to be compact, • if every open cover of X has a finite subcover. • A subset A of a space X is compact, • if it is compact as a subspace of X.

  16. Definition (Compact subset) • A space X is said to be compact, • if every open cover of X has a finite subcover. • A subset A of a space X is compact, • if it is compact as a subspace of X. • That is , if A is compact and ,

  17. Definition (Compact subset) • A space X is said to be compact, • if every open cover of X has a finite subcover. • A subset A of a space X is compact, • if it is compact as a subspace of X. • That is , if A is compact and , • Then ,

  18. Definition (Compac subset) • A space X is said to be compact, • if every open cover of X has a finite subcover. • A subset A of a space X is compact, • if it is compact as a subspace of X. • That is , if A is compact and , • Then , • Where is finite and each is open in X.

  19. Example • All finite spaces are compact and are called trivial compact spaces.

  20. Example • All finite spaces are compact and are called trivial compact spaces. • Every cofinite space is compact. (verify)

  21. Example • All finite spaces are compact and are called trivial compact spaces. • Every cofinite space is compact. (verify) • R with the usual topology is not compact.

  22. Example • All finite spaces are compact and are called trivial compact spaces. • Every cofinite space is compact. (verify) • R with the usual topology is not compact. • No infinite discrete space is compact (verify)

  23. Theorem. Any finite union of compact subsets of a space X is compact.

  24. Theorem.A continuous image of a compact space is compact.

  25. Theorem. The continuous image of compact subset A of a space X is compact.

  26. Theorem (Closed Hereditary Propety) A closed subspace of a compact space is compact.

  27. Good Luck

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