Скачать или смотреть LEC # 18|| METRIC SPACES ||SHOW THAT EMPTY SET AND FULL SPACE X ARE OPEN SET||P.U, CSIR, NET, GATE.

COURSE METRIC SPACES|| B.S MATHEMATICS|| 4TH SEMESTER|| COURSE CODE MATH-204|| PUNJAB UNIVERSITY||CSIR, NET, GATE.
LECTURE # 18|| THEOREMS RELATED TO OPEN SETS
In this lecture I shall discuss three important theorems related to open sets that can be stated as
Theorem:1 Show that in any metric space X, empty set and full space X are open sets.
Theorem:2 Show that open sets in usual metric R are open intervals.
Theorem:3 Show that every non empty subset of a discrete metric space is open.
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This course is specially designed for students of 4th semester of Punjab university Lahore Pakistan.
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COURSE CONTENTS METRIC SPACES ARE
1. Definition of metric spaces.
2. Axioms of metric spaces.
3. Usual or Euclidean Metric spaces.
4. Bounded metric.
5. Examples of metric spaces.
6. Discrete metric.
7. Pseudo metric.
8. Cauchy Schwarz Inequality.
9. Minkoski’s Inequality.
10. Open and closed Balls.
11. Examples of open and closed balls/ spheres.
12. Theorems on open and closed balls.
13. Open and closed sets.
14. Examples of open and closed sets.
15. Theorem of open and closed sets.
16. Hausdroff Axioms.
17. Neighborhood.
18. Example of neighborhoods.
19. Theorem of neighborhoods.
20. Concept of limit points.
21. Examples of limit points.
22. Theorem of limit points.
23. Closure of a set.
24. Examples and theorems of Closure of a set.
25. Interior of a set.
26. Examples and theorems of Interior of a set.
27. Exterior of a set.
28. Examples and theorems of Exterior of a set.
29. Boundary or Frontier of a set.
30. Examples and theorems of Boundary Frontier of a set.
31. Sequence in metric spaces.
32. Convergent sequence.
33. Divergent sequence.
34. Archimedean Property.
35. Complete metric spaces.
36. Cauchy sequence.
37. Continuity in metric spaces.

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