Are Finite Families of Closed Sets Closed?

[shanepitts]

Homework Statement

Let {E_i: 1≤i≤n} be a finite family of closed sets. Then ∪_i=1ⁿ E_i is closed.

Homework Equations

Noting that (E_i)^c is open

The Attempt at a Solution

Honestly, I have no idea where to start.

I tried to demonstrate that E_ai≥E_i if a is a constant greater than zero. Then showing that E_ai is closed which means that E_i is closed.

 

[FeDeX_LaTeX]

This is easiest to do if you instead try showing that, if E_i are open sets, where 1 \leqslant i \leqslant n, then \bigcap_{i=1}^{n} E_{i} is open, which pretty much follows from the definition of an open set.

You can then take complements and you're done.

 

[Fredrik]

When you have more experience with proofs, you will find it very easy to see where to start in problems like this. There's a statement P that you're supposed to use to prove a statement Q. In this situation, you should almost always ask yourself what the definitions of the terms in statement Q are telling you that statement Q means.

What does it mean to say that $\bigcup_{i=1}^n E_i$ is closed? The answer depends on your definition of closed. The most common definition is "a set is said to be closed if its complement is open". If that's your definition, the answer is: $\left(\bigcup_{i=1}^n E_i\right)^c$ is open. The proof of that is rather trivial, if you know the most basic theorems.

If you're working with metric spaces, perhaps your definition of closed is something else, e.g. "a set E is said to be closed if the limit of every convergent sequence in E is in E". Then the definition suggests another strategy: Start by saying "Let $(x_n)_{n=1}^n$ be a convergent sequence in $\bigcup_{i=1}^n E_i$". Then try to prove that $\lim_n x_n\in \bigcup_{i=1}^n E_i$.

 

[HallsofIvy]

As both FedExLatex and Fredrick said, how you prove something often depends on how it is defined! Exactly how are open and closed sets defined in your course?

 

[shanepitts]

When you have more experience with proofs, you will find it very easy to see where to start in problems like this. There's a statement P that you're supposed to use to prove a statement Q. In this situation, you should almost always ask yourself what the definitions of the terms in statement Q are telling you that statement Q means.

What does it mean to say that $\bigcup_{i=1}^n E_i$ is closed? The answer depends on your definition of closed. The most common definition is "a set is said to be closed if its complement is open". If that's your definition, the answer is: $\left(\bigcup_{i=1}^n E_i\right)^c$ is open. The proof of that is rather trivial, if you know the most basic theorems.

If you're working with metric spaces, perhaps your definition of closed is something else, e.g. "a set E is said to be closed if the limit of every convergent sequence in E is in E". Then the definition suggests another strategy: Start by saying "Let $(x_n)_{n=1}^n$ be a convergent sequence in $\bigcup_{i=1}^n E_i$". Then try to prove that $\lim_n x_n\in \bigcup_{i=1}^n E_i$.

As both FedExLatex and Fredrick said, how you prove something often depends on how it is defined! Exactly how are open and closed sets defined in your course?

Here are the definitions my professor provided us

 

[HallsofIvy]

Okay,, a set is closed if and only if its complement if open. That was the first definition Fredrick gave. I notice you did NOT define "open set". In general "point set" topology, a "topology" for a set, X is a collection of subsets of X such that, X itself is in the collection, the empty set is in the collection, the union of any number of sets in the collection is also in the collection and the intersection of any finite number of sets in the collection is also in the set. To show that "the union of a finite number of closed sets is closed", using that definition, you have to show that its complement is open. To do that you can use the fact that \left(\cup_{n} A_n\right)^c= \cap_{n}A_n^c.

 

[shanepitts]

Okay,, a set is closed if and only if its complement if open. That was the first definition Fredrick gave. I notice you did NOT define "open set". In general "point set" topology, a "topology" for a set, X is a collection of subsets of X such that, X itself is in the collection, the empty set is in the collection, the union of any number of sets in the collection is also in the collection and the intersection of any finite number of sets in the collection is also in the set. To show that "the union of a finite number of closed sets is closed", using that definition, you have to show that its complement is open. To do that you can use the fact that \left(\cup_{n} A_n\right)^c= \cap_{n}A_n^c.

Thank you, this helps a great deal

cheers

 

[Fredrik]

Since the definitions you quoted specifically mentioned $\mathbb R^n$, rather than some arbitrary topological space, it's very likely that your book's definition of open is "a set E is said to be open if every element of E is an interior point of E". This should tell you how to proceed with the proof.