- #1

Eclair_de_XII

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- 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\}##

%%%

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

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##

%%%

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\}##

%%%

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

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