What is the significance of the phrase "non empty"?

- 2bornot2b

What is the significance of the phrase "non empty"?

- chestercat

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- anonymous

It means that a set does have elements, and is not an empty set.

- 2bornot2b

I find the phrase "nonempty" absent in the definition of supremum, but it is present in the definition of completeness axiom. Can you explain why?

- 2bornot2b

Should I write down the definitions here?

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- JamesJ

I know what they are. I'm sure in the context, it's clear that the set for which you're finding the sup is not empty. If it were, the sup wouldn't be defined.

- 2bornot2b

Is it " If it were" or " If it weren't"

- 2bornot2b

I didn't get you ...

- JamesJ

If a subset of the real number, \( A \subset \mathbb{R} \), were empty, then it would have no sup.

- JamesJ

Hence
\[ \sup A \]
is defined if and only if \( \emptyset\neq A \subset \mathbb{R} \) and \( A \) has an upper bound.

- JamesJ

the least upper bound is the supremum. Different words, same thing.
Likewise for glb and inf

- JamesJ

I am. The lub = least upper bound. The lub is the supremum. Same thing.

- 2bornot2b

And why is it absent in the definition of upperbound

- JamesJ

upper bound or least upper bound?

- 2bornot2b

Just upper bound

- JamesJ

Give me the definition you're talking about. But in any case, we say the empty set has no upper or lower bound; we just don't define it if the set is empty.

- 2bornot2b

Definition of Upper Bound: Let S be a set of real numbers. If there is a real number b such that \[x\le b\] for every x in S, then b is called an upper bound for S and we say that S is bounded above by b

- 2bornot2b

My point is why not say "let S be a non empty set of real numbers"

- JamesJ

Very technically, you are correct. Because by this definition, every real number is an upper bound of the empty set. And that leads to a clash with the completeness axiom because there is no least upper bound, no sup of the empty set.
So to avoid this problem, we should say the empty set has no upper bound. agreed.

- 2bornot2b

But I copied that definition from T.M. Apostol,
of course someone before me would have pointed this out, if it were not acceptable.

- JamesJ

It could also be clear in the context in which this definition is given that the subset is non-empty.
But in any case, whatever this book is, it isn't the word of god.
Take it from me that, yes: this only makes sense if the subset is non-empty. And move on.

- 2bornot2b

Sorry my computer crashed, and it got restarted.

- 2bornot2b

Also take a look at this definition of neighbourhood of a point (Its from the lecture note of our prof)
Let \[c\in \mathbb{R}\]. A subset S of R is said to be a neighbourhood of c, if there exists an open interval (a,b) such that \[c\in (a,b)\subset S\]

- 2bornot2b

I mean I would like to state it as "Let c∈R. A non empty subset S of R is said to be a neighbourhood of c, if there exists an open interval (a,b) such that
c∈(a,b)⊂S"

- JamesJ

An open interval is by definition necessarily non-empty, so in this case, I would argue it's not necessary. To be more explicit, an open interval is a subset
\[ T = \{ x \ | \ a < x < b \} \]
where \( a < b \) and therefore \( T \) is not empty.

- 2bornot2b

Have a look at this http://en.wikipedia.org/wiki/Empty_set#Extended_real_numbers
The first line.. that starts with "since the empty set........."

- JamesJ

Right, whatever that article is talking about "Extended Reals" or whatever else you want to call it, that's not the Real Numbers as well usually define them.

- 2bornot2b

Okay,... I am getting you....
Thanks.

- JamesJ

**we [not well]

- 2bornot2b

what did you mean by that, I am sorry I didn't understand that...

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