## walters Group Title Show that {x} are open sets in X for all points x∈X, then all subsets of X are also open in X. one year ago one year ago

1. experimentX Group Title

{x} collection of open sets ... open covering of X?

2. walters Group Title

yes

3. experimentX Group Title

looks like i am not understanding the Q ... you sure the Q is right?

4. walters Group Title

yes it is like that

5. experimentX Group Title

any background on the Q? for some reason I find point set topology very hard.

6. walters Group Title

what i know that the set is open when all its point are interior

7. experimentX Group Title

But discrete points are itself closed set.

8. experimentX Group Title

where does this problem come from (book) ??

9. walters Group Title

assignment

10. experimentX Group Title

which book are you using as Text Book ??

11. walters Group Title

RG Bartle and DR Sherbert ,introduction to real analysis john wiley &sons

12. experimentX Group Title

@walters have you managed to go all though chapters to reach Point Set Topology at the end?

13. walters Group Title

not really but i think his question is open and closed sets

14. walters Group Title

i understand it this way but i can't show it |dw:1361566625628:dw|

15. walters Group Title

|dw:1361567087644:dw|

16. experimentX Group Title

Let F be collection of A's ... and $$X = \cup_{A \in F} A$$ and $$B(x;r) \subset A_i \forall x \in A, \implies B(x;r) \subset X$$ shows that X is open set.

17. experimentX Group Title

is X compact??

18. walters Group Title

"compact "what do u mean

19. experimentX Group Title

I think what you are doing is Let X be a closed set, and $$x \in X$$ be a point in X. and $$X - {x}$$ is a subset of X which is open in X. let me ask others.

20. experimentX Group Title

X-{x}

21. walters Group Title

is what i am thinking "but not quit sure"

22. experimentX Group Title

23. walters Group Title

i think since the subset of X are open this also implies that {x}and its complements are also open

24. experimentX Group Title

i got the hint: in the discret topology all points are open

25. walters Group Title

so does this mean that there is no relationship between {x} and its complement

26. experimentX Group Title

no ... use the fact that union of infinite number of open sets is and open set.

27. experimentX Group Title

or simply just union of open sets.

28. walters Group Title

i mean like "ie if {x} is open it does not imply that its complement is also closed "

29. walters Group Title

is it possible ?

30. experimentX Group Title

I guess not in discrete topology

31. experimentX Group Title

|dw:1361569963336:dw| usually we have neither closed nor open set.

32. experimentX Group Title

but in discrete topology we seem to have http://mathworld.wolfram.com/DiscreteSet.html http://mathworld.wolfram.com/DiscreteTopology.html

33. experimentX Group Title

the neighborhood of the boundary will not contain any point of that inner set, which makes the compliment open.

34. walters Group Title

ok i get ur statement

35. experimentX Group Title

as to your question, the subsets of X will be power sets of its elements which are open. And the union of open set is open. Hence all the subsets will be open.

36. experimentX Group Title

P.S. I am not sure, I don't have experience with Topology.

37. walters Group Title

@phi

38. phi Group Title

I am an engineer, and have not studied analysis.

39. walters Group Title

@JamesJ and @KingGeorge

40. walters Group Title

|dw:1361653147330:dw| i think they mean it this way because of the question

41. walters Group Title

@jacobian