Question

pleas can anyone helm me prove that arbitrary intersection of open set is not open????

Answer 1

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?

Answer 2

metric space sir

Answer 3

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

Answer 4

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

Answer 5

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.

Answer 6

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

Answer 7

I like that @Loser66. Very elegant.

Answer 8

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

Answer 9

hey @jtvatsim now is your turn :)

Answer 10

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

Answer 11

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.

Answer 12

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.

Answer 13

I think that the arbitrary union of open sets is open

Answer 14

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.

Answer 15

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

Answer 16

http://nou.edu.ng/uploads/NOUN_OCL/pdf/edited_pdf3/MTH%20301.pdf

Answer 17

any help sirs?

Answer 18

Go to this page , read section 2.1. It is wayyyyyyyyyyy clearer than your text book
https://books.google.com/books?id=mMBY5jdjGfoC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

yes sir @Loser66

Answer 23

How about this? watch section 14 topology. https://www.youtube.com/watch?v=FHL4udeLf9Q&index=14&list=PLZzHxk_TPOStgPtqRZ6KzmkUQBQ8TSWVX

Answer 24

This one is different than Loser66's but I've heard good things about this one: http://www.topologywithouttears.net/topbook.pdf

Answer 25

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

Answer 26

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

Answer 27

ok thanks @Michele_Laino

Answer 28

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