Closed Set Proof: Homework Statement

[STEMucator]

Well, we just agreed that 0 and 1 are not in the union. So the union cannot be [0,1].

What if I take x such that 0 < x < 1? Is x in the union?

Yes x would indeed be in the union then. So I'm guessing I had to be particular about the endpoints i.e (0,1)

 

[jbunniii]

Right, (0,1) is the answer. So think for a moment about why this question was asked as part (b), right after you showed that the union of a finite number of closed sets is closed.

 

[STEMucator]

Right, (0,1) is the answer. So think for a moment about why this question was asked as part (b), right after you showed that the union of a finite number of closed sets is closed.

So part (a) told me that the union of finitely many closed sets is closed. Hence finitely. Now we have an INFINITE amount of closed intervals, but their union is (0,1), a single open interval.

I'm thinking that I have to argue that this must be open because \forall x \in (0,1), \exists δ>0 | (x-δ, x+δ) \subseteq (0,1)

 

[jbunniii]

So part (a) told me that the union of finitely many closed sets is closed. Hence finitely. Now we have an INFINITE amount of closed intervals, but their union is (0,1), a single open interval.

I'm thinking that I have to argue that this must be open because \forall x \in (0,1), \exists δ>0 | (x-δ, x+δ) \subseteq (0,1)

Yes, that's right. All open intervals are open sets, for the same reason. You can always find room within the set to fit a neighborhood around any point.

 

[STEMucator]

Yes, that's right. All open intervals are open sets, for the same reason. You can always find room within the set to fit a neighborhood around any point.

Wow is that really it? Man I love the real number system... even though its a tad bit broken, but nonetheless part (a) was very convenient here.

 

[jbunniii]

I'm off for the evening. I think you have this problem wrapped up now, nice work! I'm sure this stuff will become a lot more intuitive as you get more practice.

 

[STEMucator]

I'm off for the evening. I think you have this problem wrapped up now, nice work! I'm sure this stuff will become a lot more intuitive as you get more practice.

Yeah man, you deserve mentor status for this one. A month or two and I'm sure ill have this down pat. Though I'm sure I'll be back with some interesting exercises from my prof tomorrow. Thanks for all your help.

 

[jbunniii]

Wow is that really it? Man I love the real number system... even though its a tad bit broken, but nonetheless part (a) was very convenient here.

The real number system is pretty amazing. You're just starting to scratch the surface - wait until you start playing around with the Cantor set and measure theory and Baire category and so forth. Really cool stuff, and it all builds on the basics that you're learning now.