walters
  • walters
Show that {x} are open sets in X for all points x∈X, then all subsets of X are also open in X.
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
jamiebookeater
  • jamiebookeater
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
experimentX
  • experimentX
{x} collection of open sets ... open covering of X?
walters
  • walters
yes
experimentX
  • experimentX
looks like i am not understanding the Q ... you sure the Q is right?

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

walters
  • walters
yes it is like that
experimentX
  • experimentX
any background on the Q? for some reason I find point set topology very hard.
walters
  • walters
what i know that the set is open when all its point are interior
experimentX
  • experimentX
But discrete points are itself closed set.
experimentX
  • experimentX
where does this problem come from (book) ??
walters
  • walters
assignment
experimentX
  • experimentX
which book are you using as Text Book ??
walters
  • walters
RG Bartle and DR Sherbert ,introduction to real analysis john wiley &sons
experimentX
  • experimentX
@walters have you managed to go all though chapters to reach Point Set Topology at the end?
walters
  • walters
not really but i think his question is open and closed sets
walters
  • walters
i understand it this way but i can't show it |dw:1361566625628:dw|
walters
  • walters
|dw:1361567087644:dw|
experimentX
  • 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.
experimentX
  • experimentX
is X compact??
walters
  • walters
"compact "what do u mean
experimentX
  • 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.
experimentX
  • experimentX
X-{x}
walters
  • walters
is what i am thinking "but not quit sure"
experimentX
  • experimentX
let me ask experts ... I'll reply you in sometime.
walters
  • walters
i think since the subset of X are open this also implies that {x}and its complements are also open
experimentX
  • experimentX
i got the hint: in the discret topology all points are open
walters
  • walters
so does this mean that there is no relationship between {x} and its complement
experimentX
  • experimentX
no ... use the fact that union of infinite number of open sets is and open set.
experimentX
  • experimentX
or simply just union of open sets.
walters
  • walters
i mean like "ie if {x} is open it does not imply that its complement is also closed "
walters
  • walters
is it possible ?
experimentX
  • experimentX
I guess not in discrete topology
experimentX
  • experimentX
|dw:1361569963336:dw| usually we have neither closed nor open set.
experimentX
  • experimentX
but in discrete topology we seem to have http://mathworld.wolfram.com/DiscreteSet.html http://mathworld.wolfram.com/DiscreteTopology.html
experimentX
  • experimentX
the neighborhood of the boundary will not contain any point of that inner set, which makes the compliment open.
walters
  • walters
ok i get ur statement
experimentX
  • 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.
experimentX
  • experimentX
P.S. I am not sure, I don't have experience with Topology.
walters
  • walters
@phi
phi
  • phi
I am an engineer, and have not studied analysis.
walters
  • walters
@JamesJ and @KingGeorge
walters
  • walters
|dw:1361653147330:dw| i think they mean it this way because of the question
walters
  • walters
@jacobian

Looking for something else?

Not the answer you are looking for? Search for more explanations.