## anonymous one year ago pleas can anyone helm me prove that arbitrary intersection of open set is not open????

1. jtvatsim

Are you working with topological spaces or metric spaces in this assignment? Basically, what is the definition of open set that you are to use?

2. anonymous

metric space sir

3. jtvatsim

OK. Thanks, that helps us know what to focus on.

4. jtvatsim

After consulting my textbook, it seems that all you need is a counterexample to prove this.

5. Loser66

Let A, B are open set then $$A\cup B$$ is an open set also Suppose $$A\cap B$$ is not empty consider compliment of $$A\cap B$$ that is the set of element in A or in B and we know that that set is open, hence $$A\cap B$$ is closed.

6. anonymous

ok sir but what textbook do you thick should be the best to study metic space for a dummie like me ?

7. jtvatsim

I like that @Loser66. Very elegant.

8. anonymous

@Loser66 can you be specific using the real line R here sir , please

9. Loser66

hey @jtvatsim now is your turn :)

10. anonymous

because i get more confuse trying to understand my text book.

11. jtvatsim

But sometimes the intersection of two open sets IS open. :) Consider (0,3) and (1,2). The intersection is (1,2) which is still open. It is arbitrary intersections that cause problems.

12. jtvatsim

Anyways, to answer your question @GIL.ojei I have yet to find a good introduction to metric spaces. I've struggled with almost all textbooks on the subject. But, I did find this example for the real line.

13. Michele_Laino

I think that the arbitrary union of open sets is open

14. jtvatsim

Consider the set of open intervals centered around the point 0. That is, the family (-1/i, 1/i). Then, consider the infinite intersection $\cap_{i = 1}^\infty (-1/i, 1/i)$ You will see that as this proceeds to infinity the intervals converge to (0,0) or just the single point {0}. Since a single point is not open, we have shown that an arbitrary intersection of open intervals need not be open.

15. anonymous

here is a link @Michele_Laino ,@jtvatsim and @Loser66 ,,, please hepl explain page 17 and 18(remark 1 and 2)

16. anonymous
17. anonymous

any help sirs?

18. Loser66

19. Michele_Laino

according to the observation of @jtvatsim if the result of an arbitrary intersection, for example an infinite intersection, is a one point set, then we can show that in a metric space, which is also a topological space, any one point set is a closed set

20. anonymous

@Loser66 can't find a way to read the text book

21. Loser66

oh, you are not in America. That's why you can't open that site. Am I right?

22. anonymous

yes sir @Loser66

23. Loser66

24. jtvatsim

25. anonymous

ok hard that before but he did not explain matric space in dept

26. Michele_Laino

metric space is a set equipped with a function, called distance

27. anonymous

ok thanks @Michele_Laino

28. anonymous

thanks @Loser66 and thats @jtvatsim .. you guys have been helpful