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Topic: help with misprint (?) Counterexamples in topology
Replies: 9   Last Post: Feb 15, 2013 8:54 AM

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William Elliot

Posts: 1,244
Registered: 1/8/12
Re: Minimal Hausdorff Topology
Posted: Feb 11, 2013 4:28 AM
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On Mon, 11 Feb 2013, fom wrote:
> On 2/11/2013 2:08 AM, William Elliot wrote:
> >
> > > I am wondering about diagram for
> > > example 100, Minimal Hausdorff Topology.

> > From what book?
> >

> > > The "special" basis neighborhoods in the diagram do not seem to
> > > coincide with the defining conditions.
> > >
> > > The second coordinate has j>n for that n indexing the basis
> > > neighborhood, but the diagram seems to indicate j<n.

> >
> > It does not.
> >

> > > Also, if I am correct about the error in the diagram, then should
> > > not the inequality be changed to j>=n so that there is a basis
> > > neighborhood associated with the first row of the diagram as with
> > > every other row?
> > >

> > No, it makes no difference if it's j >= n or j > n.
> >

> > > I am probably just misreading the
> > > definition. But it has me confused.
> > >

> > It takes some study to understand.
> >
> > There are orders, that of A and that of Z+. The order for Z+ begins at
> > the bottom with 1 and goes to the top. Notice how at the top the rows
> > of dots are closer that at the bottom indicating a two dimensional
> > dot, dot, dot. for the rows.

>
> Thanks. I simply inverted the order for Z+
> Now it makes sense.
>
> If you want to see how I am trying to understand how to use it, open the
> post on "distinguishability"
>

Two points a,b, of topological space are indistinguishable when
for all open U, a in U iff b in U.

Thusly, tow points a,b of space are topologically distinguishable
iff there's some open U with a in U, b not in U or a not in U, b in U.

A T0 space is a space in which every two distinct points are topologically
distinguishable. All the points of an indiscrete space are topologically
indistinguishable.

Two topological spaces are topologically indistinguishable when they're
homeomorphism. Two groups or rings are algebraically indistinguishable
when they're isomomorphic.

> The post is way too long, but the attempted application of the topology
> is at the very end.


Yes, way to long, detailed, rambling and pointless.



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