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

2. 2 Neighborhood of a point x in E 1 ･ x-rx-r x x+rx+r N E1E1

3. 3 Any subset containing a neighborhood is another neighborhood. ･ x-rx-r x x+rx+r N E1E1 N１N１

4. 4 Accumulation Points

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

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

7. 7 Derived Set

8. 8 Limits of Sequences

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

10. 10 1.1 Prop. A convergent sequence in E 1 has a unique limit. (()) Suppose we have two limits x and y. We can separate them by some of their neighbors as shown below. xy I J

11. 11 1.2 Monotonic Limits Theorem

12. 12 Cauchy sequence

13. 13 1.3 Convergence Characterization

14. 14 Accumulation and Convergence

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

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

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

18. 18 ･ x ･ x ･ x r r r N N N Neighborhood in E n of a point x

19. 19 2.1 Neighborhood property

20. 20 Open set

21. 21 Closed set

22. 22 Propositions 1.A subset is open in E n if and only if its complement is closed in E n. 2.Any union of open sets is open. 3.Any intersection of closed sets is closed.

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

24. 24 Any union of open sets is open.

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

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

27. 27 Metric subspaces

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

29. 29 Proof of Th. 3.5(b)

30. 30 Closure Omitted

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

32. 32 Pinching is continuous. ･ ･

33. 33 Gluing is continuous

34. 34 4.1 Continuity Characterization