fom
Posts:
1,031
Registered:
12/4/12
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Re: Minimal Hausdorff Topology
Posted:
Feb 11, 2013 10:38 PM
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On 2/11/2013 3:28 AM, William Elliot wrote:
>>
> Two points a,b, of topological space are indistinguishable when
> for all open U, a in U iff b in U.
Once again, thank you.
As I mentioned before, I have seen you making
a number of posts with set-theoretic problems.
You may be unaware that identity in set theory
is not the same as what you have described with
your explanation above.
You can get the explanation of identity that
does apply to set theory at
http://plato.stanford.edu/entries/identity-relative/#1
To see what I mean, this is how Kunen puts the
matter:
"Intuitively, x=y means that x and y
are the same object. This is reflected
formally in the fact that the basic
properties of equality are logically
valid and need not be stated explicitly
as axioms of ZFC. For example,
|- (x=y -> Az(zex <-> zey))
whereas the converse is not logically
valid, [...]"
If you look this up, you will feel that
I have excluded something important. But,
the rest of the statement involves a
theorem based on extensionality and
pairing.
Jech begins with the statement of
extensionality followed by the
disclaimer:
"If X and Y have the same elements
then X=Y:
Au(ueX <-> ueY) -> X=Y
The converse, namely, if X=Y then
ueX <-> ueY, is an axiom of predicate
logic."
So, there are reasons why I took such
care in the construction.
At the end, the object had been to
topologize the structures so that the
sense of what you have written above
was explicitly demonstrated as being
essential to the construction of the
reals by Dedekind cuts.
Since the explanation of Leibniz'
Law in the article cited above is
not what Leibniz actually wrote when
explaining it, the proper way of
introducing a metric structure is
found in Kelley's discussion of uniform
spaces and his metrization lemma.
Given a "logical" identity, the
psuedometric axiom
x=y -> d(x,y)=0
puts a topology onto it.
And, since you are so competent when it
comes to topology, go to the back of
Steen and Seebach to the section where
they discuss non-metrizable topologies:
Tangent Disc Topology
(and Bing's flow space variation)
Tangent Disc Subspaces
Cantor Tree
Moore's Road Space
And, consider Fleissner's example
mentioned in the Addendum to that
section in the context of
http://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry
I appear inept only because I am
out of practice. I work on very
hard problems -- mostly as a pastime
on my way to and from a sometimes
fun but generally menial job.
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