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son
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Prove that the intersection of any collection of closed sets in a
topological space X is closed.
topological space X is closed.
son said:the theorem i am using for a closed set is...
Let X be a topological space. the following statements about the collection of closed set in X hold:
(i) the empty set and X are closed
(ii) the intersection of any collection of closed sets is a closed set
(iii) the union of finitely many closed sets is a closed set
I would start it something like:son said:the definition of a closed set is... a subset A of a topological space X is closed if the set X - A is open.
but I'm not sure how i would start the proof...
No, you cannot. That works only in a "metric space" because balls are only defined in a metric space. This problem clearly is about general topological spaces.culturedmath said:I am new in the forum ( although I have read it for some time ) and I am not quite sure how much of a hint I am allowed to give you but:
You can prove that a set is closed using "balls". I would suggest you to work in this direction.
A closed set in a topological space is a subset of the space that contains all of its limit points. In other words, a closed set includes all of the points that can be approached arbitrarily closely by points in the set itself.
The intersection of two closed sets in a topological space is a new set that contains only the points that are common to both of the original sets. This new set is also a closed set, as it contains all of the limit points of the original sets.
Yes, the intersection of any collection of closed sets in a topological space is always a closed set. This is because the intersection contains only the points that are common to all of the original closed sets, making it a closed set itself.
The intersection of any collection of closed sets in a topological space is a subset of the closure of that same set. In other words, the intersection is a "subset" of the closure, meaning it contains some, but not necessarily all, of the points in the closure.
Yes, it is possible for the intersection of closed sets in a topological space to be empty. This occurs when there are no points that are common to all of the original closed sets, meaning the intersection contains no points and is therefore an empty set.