Question

Let \[K_1,K_2,...,K_N\] be a compact subsets of the metric space (X,d). Show that

a) \[K_1\cup K_2\cup...\cup K_N\] is compact.

b) \[K_1∩K_2∩...∩K_N\] is compact

Answer 1

Hint: I have to use the Pigeonhole principle. But I am not sure how to show this.

Answer 2

I'm sorry I don't know the answer. I have abandoned the study of higher mathematics, since I make research on nuclear physics

Answer 3

sorry, I don't know either.

Answer 4

Since no one has answered, I will give you my thoughts. It seems like you are trying to prove that the unions and intersections of compact sets are compact.

Take this explanation of the pigeon hole principle "In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. This theorem is exemplified in real-life by truisms like "there must be at least two left gloves or two right gloves in a group of three glove"

In this case, I would say we have n sets. Take the contrapositive if it is not a compact set and show by contradiction that the union of compact sets must be compact.

That is my best guess, I hope it helps till someone else can answer.

Answer 5

To be dead honest, openstudy typically targets K-12 or K-14 education... I would suggest posting on stack exchange network (math.stackexchange; make sure if you are new to the site you read the posting rules) for a larger body of capable answerers. We do have a few people here that are familiar with higher math such as @Kainui and @ganeshie8 , but they're of course not always online. Best of luck!

Answer 6

You can use induction if you can prove it for the union/intersection for any two K sets.

Answer 7

Just remember to lists the properties of a compactness, then it shouldn't be too hard.