Metric Topology

1. Metric Topology http://cis.k.hosei.ac.jp/~yukita/

2. Neighborhood of a point x in E1 x+r x-r x ･ E1 N

3. Any subset containing a neighborhood is another neighborhood. x+r x-r x ･ E1 N N１

4. Accumulation Points

5. The open interval (a,b) accumulates at each a<x<b. x+r x-r x b a x b x+r x-r a ･ ･ m M m M whatever is the case x+r b x-r x-r x+r x b a a x ･ ･ m M m M

6. The closed interval [a,b] accumulates at each abxbb. whatever is the case x+r x-r x b a x b x+r x-r a ･ ･ m M m M x+r b x-r x-r x+r x b a a x ･ ･ m M m M x-r x+r x-r x+r a b x=b x=a ･ ･ m M m M

7. Derived Set

8. Limits of Sequences

9. Limits of Sequences (Ex12,p.45)

10. 1.1 Prop. A convergent sequence in E1 has a unique limit. Suppose we have two limits xand y. We can separate them by some of their neighbors as shown below. I J ( ) ( ) x y

11. 1.2 Monotonic Limits Theorem

12. Cauchy sequence

13. 1.3 Convergence Characterization

14. Accumulation and Convergence

15. 1.4 Limit-Accumulation Properties To be filled in the future.

16. Open n-ball about x with radius r r ･ r ･ x x ･ x r

17. Closed n-ball about x with radius r r ･ r ･ x x ･ x r

18. Neighborhood in En of a point x N r ･ r ･ x x ･ x N N r

19. 2.1 Neighborhood property

20. Open set

21. Closed set

22. Propositions • A subset is open in En if and only if its complement is closed in En. • Any union of open sets is open. • Any intersection of closed sets is closed.

23. A subset is open in En if and only if its complement is closed in En. U F

24. Any union of open sets is open.

25. Any intersection of closed sets is closed.The dual of the previous proposition

26. An open set is a union of open balls.

27. Metric subspaces

28. Theorem 3.5 Notice that (a) is a special case of (b).

29. Proof of Th. 3.5(b)

30. Closure • Omitted

31. Continuity f(x) f x f(A) A This kind of situation violates the condition.

32. Pinching is continuous. ･ ･

33. Gluing is continuous

34. 4.1 Continuity Characterization