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## walters 2 years ago Show that {x} are open sets in X for all points x∈X, then all subsets of X are also open in X.

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1. experimentX

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

2. walters

yes

3. experimentX

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

4. walters

yes it is like that

5. experimentX

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

6. walters

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

7. experimentX

But discrete points are itself closed set.

8. experimentX

where does this problem come from (book) ??

9. walters

assignment

10. experimentX

which book are you using as Text Book ??

11. walters

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

12. experimentX

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

13. walters

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

14. walters

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

15. walters

|dw:1361567087644:dw|

16. experimentX

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

is X compact??

18. walters

"compact "what do u mean

19. experimentX

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

X-{x}

21. walters

is what i am thinking "but not quit sure"

22. experimentX

let me ask experts ... I'll reply you in sometime.

23. walters

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

24. experimentX

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

25. walters

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

26. experimentX

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

27. experimentX

or simply just union of open sets.

28. walters

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

29. walters

is it possible ?

30. experimentX

I guess not in discrete topology

31. experimentX

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

32. experimentX

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

33. experimentX

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

34. walters

ok i get ur statement

35. experimentX

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

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

37. walters

@phi

38. phi

I am an engineer, and have not studied analysis.

39. walters

@JamesJ and @KingGeorge

40. walters

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

41. walters

@jacobian

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