The set of the real numbers is closed

[tom.stoer]

The set of the real numbers is closed.

For me this is nearly trivial (*) but perhaps I miss something; a colleagues insists that there are some deeper considerations why this is far from trivial - but I don't get his point

(*)
A) A set is closed if its complement is open; the complement of R is ø which is open, therefore R is closed
B) In a topological space, a set is closed if it contains all its limit points; this applies directly to R (see also D)
C) In a complete metric space, a set is closed if it's constructed as the closure w.r.t. to its limit operation; the set of real numbers can be constructed as completion of the rational numbers in the sense of equivalence classes of Cauchy sequences; then Q is dense in R by construction and R is closed by construction

Do I miss something? Are there more fundamental definitions of closed sets? Is it problematic that closed sets are defined via limiting points of convergend sequences whereas this does not apply to R as a whole b/c it misses the "boundary" of R which could be defined by divergent sequences (x_n) → ∞?

I do not see such problems, but perhaps I am missing something.

 

[WannabeNewton]

There are no problems. Most of the definitions you have given are derived equivalent definitions of 'closed' from the topological standpoint wherein the most fundamental definition of being closed is simply that of a set whose complement is open. In topology we take by definition that both the overarching space and the empty set belong to the topology so the overarching space is always both open and closed. There's nothing deep about this, it is trivial like you said.

Every element of the reals is the limit of a sequence of rationals as per the Cauchy construction so there's no problem there either if you want to look at things from the less fundamental perspective of metric spaces. $\partial \mathbb{R} = \varnothing$ which is exactly what characterizes the fact that $\mathbb{R}$ is both open and closed in its topology.

 

[tom.stoer]

thx!

 

[AlephZero]

Maybe the point of such an "almost trivial" question was to show that a naive idea like "a closed set is something like a closed interval that includes its end points, and an open set is something like an open interval that doesn't include its end points" is wrong.

 

[PeroK]

As mentioned above, as a metric space R is closed. But, perhaps your colleague is alluding to the development of R as the closure of the rationals, which is far from trivial if you do it rigorously.

In other words, it's not that easy to justify rigorously a continuous number line.

 

[tom.stoer]

thx for your ideas; unfortunately nothing like that does apply; my conclusion is that he got it wrong ;-)

 

[jgens]

But, perhaps your colleague is alluding to the development of R as the closure of the rationals, which is far from trivial if you do it rigorously.

This is nitpicking: That statement should probably read something like "[...] alluding to the development of R as a completion of the rationals [...]" since there are multiple completions of the rationals.

 

[D H]

The empty set is clopen, not open, and R is similarly clopen rather than closed. Both sets have no boundary points, and thus they simultaneously contain all and none of their boundary points.

 

[R136a1]

This is nitpicking: That statement should probably read something like "[...] alluding to the development of R as a completion of the rationals [...]" since there are multiple completions of the rationals.

Well, if we're going to nitpick: the statement should read "$\mathbb{R}$ with the Euclidean metric is a completion of the rationals" Since there are multiple topologies on $\mathbb{R}$

 

[R136a1]

The empty set is clopen, not open, and R is similarly clopen rather than closed. Both sets have no boundary points, and thus they simultaneously contain all and none of their boundary points.

How is the empty set not open?

 

[D H]

How is the empty set not open?

How is the empty set not closed?

It's both open and closed, or clopen.

 

[R136a1]

How is the empty set not closed?

It's both open and closed, or clopen.

Yes, it is closed and it is open. Therefore I found it a bit weird that you said that the empty set is not open.

 

[tom.stoer]

The empty set is clopen, not open, and R is similarly clopen rather than closed.

nitpicking again, but the statement "the empty SRT is clopen, not open" is logically wrong

a set is clopen iff it is open and closed; R is clopen; therefore R is closed and open; therefore R is closed

a number is preven iff it is prime and even; 2 is preven; therefore 2 is prime and even; therefore 2 is prime

The statement "2 is preven, not prime" is more than weird.

 

[WWGD]

There are no problems. Most of the definitions you have given are derived equivalent definitions of 'closed' from the topological standpoint wherein the most fundamental definition of being closed is simply that of a set whose complement is open. In topology we take by definition that both the overarching space and the empty set belong to the topology so the overarching space is always both open and closed. There's nothing deep about this, it is trivial like you said.

Every element of the reals is the limit of a sequence of rationals as per the Cauchy construction so there's no problem there either if you want to look at things from the less fundamental perspective of metric spaces. $\partial \mathbb{R} = \varnothing$ which is exactly what characterizes the fact that $\mathbb{R}$ is both open and closed in its topology.

Every topological space is both closed and open in its topology.

 

[1MileCrash]

The empty set is clopen, not open, and R is similarly clopen rather than closed.

Saying that R is "clopen rather than closed" seems to imply some sort of dichotomy between the two. Closed doesn't mean "closed but not open," it means closed. It is never wrong to call a clopen set closed. This is like saying that 3 is an integer "rather than" a real.