How Do Topological Closures Relate in Union Operations?

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SUMMARY

The closure of the union of two non-closed subsets A and B of a topological space X is equal to the closure of A union the closure of B, formally expressed as Cl(A ∪ B) = Cl(A) ∪ Cl(B). This relationship is fundamental in topology and can be proven by substituting definitions related to closures and unions. Additionally, for set theory, the equivalence A = B is defined as A ⊆ B and B ⊆ A, emphasizing the importance of understanding definitions in mathematical proofs.

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If A and B are two non-closed subsets of X, how would one prove that the closure of A union B= the closure of A union closure of B? Also, what site would you recommend I download TeX from when I get my new computer (Dell, runs on windows)?
 
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A great many problems are trivial if you just restate the problem by substituting in definitions.


And don't forget that, for sets, A = B means the same thing as [itex]A \subseteq B \wedge B \subseteq A[/itex].



If you have trouble substituting definitions, then this will be a weakness that will cause problems for a long time, so you should work on it specifically. I make a big point of this because it's easy to think that you're just having trouble with the subject material and never remedy the problem... and then it crops up in the next subject, and the next... if it doesn't apply to you, then ignore this paragraph. :smile:
 
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