ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:

sniping and switching the order to avoid insanity

I mean that A is in the topology associated with D.
This is the case if and only if for each x in A there is
e(x)>0 such that tHe neighborhood
{y: D(x,y) < e(x) } is a subset of A.

Ah, now i see new things. In this case, i'd like to bother you
with the following two questions. :)

You said that:

The open sets of the d1 topology are unions of open (in [0,1])
intervals. For example, (0,.3) U (.4,, is an open subset of
[0,1], and so is [0,.3) U (.4,..

Is the above, namely that ]0,1/3[ as well as [0,1/3[ being
open subsets of [0,1] , true because of the fact that we can
unify our way down to zero as follows?

lim N->oo ( U( ]0+1/i;1/3[ , i=4 , N ) )

Yes, this representation of ]0,1/3[ shows that it is open.

In that case we're agreed that ]0,1/3[ is open. However,
you claimed also that [0,1/3[ is open too. How can one show
it? The rest we've got but this part still remains unlcear.

By the way, we're working under the assumption that the
first identity doesn't hold and the second does.

U( ]a+1/i;b-1/i[ , i=1,...,n ) == U( [a+1/i;b-1/i] , i=1,...,n )
U( ]a+1/i;b-1/i[ , i=1,...,oo ) == U( [a+1/i;b-1/i] , i=1,...,oo )
where a and b are suitably chosen (say a=1 and b=1000000,
if one wishes to be picky, let's wish not to be picky, hehe)

lim N->oo ( U( ]1/2+1/i;3/4[ , i=7 , N ) )
hence creating a half-closed interval [1/2;3/4[ ?

But this union, namely
U{ ]1/2+1/i, 3/4[ , i=7,8,...},
is the open interval ]1/2,3/4[.

Thanks for the correction.

ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:

sniping and switching the order to avoid insanity

I mean that A is in the topology associated with D.
This is the case if and only if for each x in A there is
e(x)>0 such that tHe neighborhood
{y: D(x,y) < e(x) } is a subset of A.

Ah, now i see new things. In this case, i'd like to bother you
with the following two questions. :)

You said that:

The open sets of the d1 topology are unions of open (in [0,1])
intervals. For example, (0,.3) U (.4,, is an open subset of
[0,1], and so is [0,.3) U (.4,..

Is the above, namely that ]0,1/3[ as well as [0,1/3[ being
open subsets of [0,1] , true because of the fact that we can
unify our way down to zero as follows?

lim N->oo ( U( ]0+1/i;1/3[ , i=4 , N ) )

Yes, this representation of ]0,1/3[ shows that it is open.

In that case we're agreed that ]0,1/3[ is open. However,
you claimed also that [0,1/3[ is open too. How can one show
it? The rest we've got but this part still remains unlcear.
[...]

Konrad Viltersten wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:

snipping and switching the order to avoid insanity
The open sets of the d1 topology are unions of open (in [0,1])
intervals. For example, (0,.3) U (.4,, is an open subset of
[0,1], and so is [0,.3) U (.4,..

Is the above, namely that ]0,1/3[ as well as [0,1/3[ being
open subsets of [0,1] , true because of the fact that we can
unify our way down to zero as follows?

lim N->oo ( U( ]0+1/i;1/3[ , i=4 , N ) )

Yes, this representation of ]0,1/3[ shows that it is open.

In that case we're agreed that ]0,1/3[ is open. However,
you claimed also that [0,1/3[ is open too. How can one show
it? The rest we've got but this part still remains unlcear.

The interval D=[0,1] has a metric d1, where d1(x,y) = |x-y|. This gives D
a topology based on the d1 metric.

Then {p in D such that d1(0,p) < 1/3} is an open ball (in D)
of radius 1/3 around 0.

So {p in [0,1] such that |p-0| < 1/3} is open.

{p in [0,1] such that |p-0| < 1/3} =
{p in [0,1] such that |p| < 1/3} =
{p in [0,1] such that p<1/3} =
[0,1/3[

David Bernier wrote/skrev/kaita/popisal/schreibt :
Konrad Viltersten wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:

snipping and switching the order to avoid insanity
The open sets of the d1 topology are unions of open (in [0,1])
intervals. For example, (0,.3) U (.4,, is an open subset of
[0,1], and so is [0,.3) U (.4,..
Is the above, namely that ]0,1/3[ as well as [0,1/3[ being
open subsets of [0,1] , true because of the fact that we can
unify our way down to zero as follows?

lim N->oo ( U( ]0+1/i;1/3[ , i=4 , N ) )
Yes, this representation of ]0,1/3[ shows that it is open.
In that case we're agreed that ]0,1/3[ is open. However,
you claimed also that [0,1/3[ is open too. How can one show
it? The rest we've got but this part still remains unlcear.
The interval D=[0,1] has a metric d1, where d1(x,y) = |x-y|. This gives D
a topology based on the d1 metric.

Then {p in D such that d1(0,p) < 1/3} is an open ball (in D)
of radius 1/3 around 0.

This is confusing me. I get that the set you described, i.e.
C := {p in D: d1(0,p) < 1/3}
is an open set. No doubt about that. But when you claim it
to be an open set in D, then we have a problem. The way i
grasp it, saying "open set in D" is equivalent to saying
"open set when intersected by D". And then, it's hard to
view it as open.

Now, i _DO_ understand that d1(p,0) = 0 < 1/3 when p
is zero itself (of course being in D still). However, the
interval we then create is [0;1/3[ and this looks to me
as left-closed-right-open interval. Just because we can
have an open ball around the point zero, it doesn't
change anything (according to my, probably inferior,
logic) because much of the open ball is outside D, no
matter how small radius we pick.

What do i miss?

So {p in [0,1] such that |p-0| < 1/3} is open.

Assuming the above i understand that:

{p in [0,1] such that |p-0| < 1/3} =
{p in [0,1] such that |p| < 1/3} =
{p in [0,1] such that p<1/3} =
[0,1/3[

And this is hence perfectly graspable. Thanks!

The Cool Giraffe wrote:
David Bernier wrote/skrev/kaita/popisal/schreibt :
Konrad Viltersten wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:
ArtflDodgr wrote/skrev/kaita/popisal/schreibt :
"The Cool Giraffe" <giraffen@viltersten.com> wrote:

snipping and switching the order to avoid insanity
The open sets of the d1 topology are unions of open (in [0,1])
intervals. For example, (0,.3) U (.4,, is an open subset of
[0,1], and so is [0,.3) U (.4,..
Is the above, namely that ]0,1/3[ as well as [0,1/3[ being
open subsets of [0,1] , true because of the fact that
we can unify our way down to zero as follows?

lim N->oo ( U( ]0+1/i;1/3[ , i=4 , N ) )
Yes, this representation of ]0,1/3[ shows that it is open.
In that case we're agreed that ]0,1/3[ is open. However,
you claimed also that [0,1/3[ is open too. How can one show
it? The rest we've got but this part still remains unlcear.
The interval D=[0,1] has a metric d1, where d1(x,y) = |x-y|. This
gives D a topology based on the d1 metric.

Then {p in D such that d1(0,p) < 1/3} is an open ball (in D)
of radius 1/3 around 0.

This is confusing me. I get that the set you described, i.e.
C := {p in D: d1(0,p) < 1/3}
is an open set. No doubt about that. But when you claim it
to be an open set in D, then we have a problem. The way i
grasp it, saying "open set in D" is equivalent to saying
"open set when intersected by D". And then, it's hard to
view it as open.

Now, i _DO_ understand that d1(p,0) = 0 < 1/3 when p
is zero itself (of course being in D still). However, the
interval we then create is [0;1/3[ and this looks to me
as left-closed-right-open interval. Just because we can
have an open ball around the point zero, it doesn't
change anything (according to my, probably inferior,
logic) because much of the open ball is outside D, no
matter how small radius we pick.

What do i miss?

That the open ball of radius r around 0 depends on the metric space
to which 0 belongs. Usually, the metric space X, Y, etc.
is stated explicitly once in a discussion, proof, exercise;
after the metric space has been stated explicitly, the
words "open ball centered at 0 of radius 1/3" are enough
so as to not get confused once someone has had practice.
So if the metric space is called X,
"open ball centered at 0 of radius 1/3" is short for:
"the set of all points in X whose distance from 0 is less than 1/3".

So if we look at the metric space Z of all integers, with the d1
metric, the open ball of radius 1/3 around 0 is {0}. Then,
{1} , {2}, {3} are also one-element open sets for the metric
space Z of the integers. Since an arbitrary union of open
sets is open, { 1, 23, 56, 90000} is also open, in the metric
space Z. In fact, any subset of Z is open in the metric space
Z.

This may sound strange, how being an open set depends on the
metric space under consideration, and all I can say is that
these definitions in topology weren't arrived at in a single
day, from the beginnings of topology.