Is [0, infinity) a closed set?

[michonamona]

Homework Statement

Is [0, infinity) a closed set?

Homework Equations

N/A

The Attempt at a Solution

It's easy to say that its not. But the solution in my textbook suggests otherwise. Why is this so?

Thanks!
M

 

[LCKurtz]

If it isn't closed, you should be able to find a limit point that isn't in the interval, right? What point would that be?

 

[vela]

Is its complement open?

 

[michonamona]

LCKurtz:

Ok, I think I understand it for integers. Say, [1,5), where the limit point 5 is not in the interval. But what confuses me is infinity. How is it precisely defined with respect to actual numbers like integers?

vela:

What is its complement? R\[0, infinity)?

Hmm...

 

[vela]

Yes, that's the complement.

Just curious, what's the definition of a closed set that your class uses?

 

[michonamona]

If its complement is open, then I know that the interval is closed. But why? why is its complement open?

I appreciate you guys for not giving me the answers directly. I'm really trying hard to understand it.

M

 

[vela]

What's the definition of an open set?

 

[michonamona]

A set that contains only its interior points...

Its boundary is contained in its complement...

 

[vela]

The complement of $[0,\infty)$ is $(-\infty,0)$. Let $x\in(-\infty,0)$. Is there an open interval centered on x that's contained in $(-\infty,0)$?

 

[chr0nicbudd]

i think that it needs to be open on both sides.
if it were (0, infinity) it would be open. but since its its [0, infinity), it is a closed interval since you could make x = 0 and have a point, y, that lied to the left of 0, yet still within the Br(0) (ball of radius r about 0).

 

[LCKurtz]

LCKurtz:

But what confuses me is infinity. How is it precisely defined with respect to actual numbers like integers?

Infinity is not part of the real numbers. [0,oo) is just a convenient alternative notation for $x \ge 0$. You might think of the symbol as indicating the values of x are not bounded above.

 

[Hurkyl]

Homework Statement

Is [0, infinity) a closed set?

Just FYI -- "closed" only makes sense relative to a space. The right question is

Is [0, +infinity) a closed subset of the reals​

But what confuses me is infinity.

It seems to be the habit to introduce it as a sort of "useful fiction". The set of nonnegative reals is quite interval-like, and it's useful to introduce a formal interval-like notation to write such things. Thus "[0, +infinity)".


However, if you go on, you should learn about the extended real numbers which are, IMO, a much better way to go about doing things. This number system has two additional numbers that the real numbers don't: -infinity and +infinity. And it turns out that every "useful fiction" that you learn in the introductory classes turns out to port to ordinary things in this more sophisticated approach, despite having the exact same notation. e.g. In the extended real numbers, [0,+infinity) is a perfectly ordinary interval, which consists of the nonnegative real numbers. (And the interval [0,+infinity] would consist of all nonnegative extended real numbers)

 

[bennyska]

It's like putting too much air in a balloon!