How Do Topological Closures Relate in Union Operations?

[Long2024]

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)?

 

[Hurkyl]

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 $A \subseteq B \wedge B \subseteq A$.



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.

 

[quicknote]

I would approach this question by first defining what a closure is. In topology, the closure of a set A is defined as the smallest closed set that contains A. This means that the closure of A includes all the points in A as well as any limit points of A. Similarly, the closure of B includes all the points in B as well as any limit points of B.

Now, to prove that the closure of A union B is equal to the closure of A union the closure of B, we can use the fact that the closure of a union of sets is equal to the union of their closures. This can be shown using the definition of closure and the fact that the closure of a set is closed.

To prove this, we can start with the closure of A union B, which is the smallest closed set that contains A union B. This means that it contains all the points in A union B as well as any limit points of A union B. Now, we can break down A union B into two sets: A and B. Since A and B are not closed, their closures will be larger than A and B themselves. Therefore, the closure of A union B will also contain the closure of A and the closure of B.

On the other hand, the closure of A union the closure of B is the smallest closed set that contains A union the closure of B. This means that it contains all the points in A union the closure of B as well as any limit points of A union the closure of B. Since the closure of B already contains all the points in B and any limit points of B, the closure of A union the closure of B will also contain A and any limit points of A. Therefore, the union of the closures of A and the closures of B will be equal to the closure of A union B.

In conclusion, we can prove that the closure of A union B is equal to the closure of A union the closure of B by using the definition of closure and the fact that the closure of a union of sets is equal to the union of their closures. As for downloading TeX on your new computer, I would recommend going to the official TeX website (https://www.tug.org/texlive/) and downloading the appropriate version for Windows.