# Proving intersection of finitely many open sets is open

• Eclair_de_XII
In summary, a collection of open sets denoted as ##P_i##, ##1\leq i\leq N## where ##N\in \mathbb{Z}^+## is defined. If ##x\in\cap_{i=1}^N P_i##, then ##x## belongs to every ##P_i##. By taking the finite intersection of these sets, we can find a smallest positive number ##r## such that ##B(x,r)## is contained in every ##P_i##. This method fails for an infinite number of sets where ##\inf\{\rho_i\} = 0##, but if ##\inf\{\rho_i\} > 0##
Eclair_de_XII
Homework Statement
Prompt in topic title. Proof describes a method in order to find an open ball centered around a point in the intersection. I assert the method will not work if there are infinitely many open sets in the intersection, which will not exclude the possibility that it will work.
Relevant Equations
Ball: A ball centered around some ##x## with radius ##r## is defined to be the set ##B(x,r):=\{y:|x-y|<r\}##.
Open: A set ##A## is open if for all ##x\in A##, there is a positive number ##r## such that ##B(x,r)\subset A##.
Define a collection of open sets to be denoted as ##P_i##, ##1\leq i\leq N## where ##N\in \mathbb{Z}^+##.

Let ##x\in\cap_{i=1}^N P_i##. By definition, ##x## must belong to every single ##P_i##.

In particular, ##x\in P_1## and ##x\in P_2##. Since ##P_1## and ##P_2## are open, there exist positive numbers ##r_1## and ##r_2## such that ##B(x,r_1)\subset P_1## and ##B(x,r_2)\in P_2##. Choose ##\inf\{r_2,r_1\}## and call it ##\rho_1##. We prove ##B(x,\rho_1)\in P_1\cap P_2##.

Let ##y\in B(x,\rho_1)##. Then ##|x-y|<\rho_1\leq r_1,r_2##. Hence,

##B(x,\rho_1)\subset B(x,r_1)\in P_1##
##B(x,\rho_1)\subset B(x,r_2)\in P_2##

By definition, ##B(x,\rho_1)\subset P_1\cap P_2##.

Now consider ##P_3##; there is ##r_3>0## such that ##B(x,r_3)\subset P_3##. Define ##\rho_2:=\inf\{\rho_2,r_3\}##. Then by similar reasoning above (in other words, change some variable names) ##B(x,\rho_2)## is contained in ##P_3## in addition to ##P_1\cap P_2##. Note that ##\rho_2=\inf\{r_1,r_2,r_3\}##

Continuing in this fashion, we obtain a decreasing sequence of ##\rho_i##, where ##1\leq i\leq N-1##. Suppose ##P_{N+1}## is an open set that ##x## also belongs to. Then there is ##r_{N+1}>0## such that ##B(x,r_{N+1})\subset P_{N+1}##. Set ##\rho_N=r_{N+1}## and ##r=\inf\{\rho_i\}_{i\in\mathbb{N}\cap [1,N]}##. By similar reasoning as above, ##B(x,r)\subset \cap_{i=1}^N P_i## and ##B(x,r)\subset P_{N+1}## which gives us our result.

Note that this choice of ##r## is invalid if there are infinitely many ##P_i## in the intersection, since in this case:

##\inf\{\rho_i\}_{i\in\mathbb{N}\cap [1,N]}=0##

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Example of intersection of infinitely many open sets being open:

##\cap_{n\in\mathbb{Z}^+}(-n,n)=(1,1)##

Note: The method above would fail because the sequence ##\rho_i## in this case would be constant at ##\rho_i=1##. So I guess, the method would fail only if the sequence ##\rho_i## were strictly decreasing.

Example of intersection of infinitely many open sets not being open:

##\cap_{n\in\mathbb{Z}^+}(-\frac{1}{n},\frac{1}{n})=\{0\}##

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I'm sure there is a less notation-heavy way of expressing this proof.

Last edited:
Your method works, but is cumbersome. You can just take finite intersection directly and apply the same reasoning. There is no need to take two sets at a time.

The proof works for a finite number of sets because a finite set of positive numbers has a smallest number. It fails for an infinite set because an infinite set of positive numbers can have inf = 0. However, if inf > 0 the intersection is open.

## 1. What is the definition of an open set?

An open set is a subset of a topological space in which every point has a neighborhood contained entirely within the set.

## 2. How do you prove that the intersection of finitely many open sets is open?

To prove that the intersection of finitely many open sets is open, we must show that for any point in the intersection, there exists a neighborhood of that point that is also contained within the intersection.

## 3. Can you provide an example of the intersection of finitely many open sets being open?

Yes, for example, consider the sets (0,2) and (1,3) on the real number line. The intersection of these two sets is the open interval (1,2), which is also open.

## 4. Are there any cases where the intersection of finitely many open sets is not open?

Yes, if the sets have empty intersection or if the intersection is a single point, then the intersection would not be considered open.

## 5. How does this concept relate to the concept of a topological space?

The concept of proving the intersection of finitely many open sets is open is important in topology because it helps define the properties of a topological space and is used in various proofs and theorems in the field.

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