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Topic: Nhood Space
Replies: 24   Last Post: Jul 3, 2013 10:43 PM

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William Elliot

Posts: 1,553
Registered: 1/8/12
Nhood Space
Posted: Jul 2, 2013 1:35 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Mon, 1 Jul 2013, Peter Percival wrote:
> Peter Percival wrote:
> > William Elliot wrote:

> > > (S,<<) is a nhood space when << is a binary relation for P(S) and
> > > for all A,B,C subset S
> > > empty set << A << S
> > > A << B implies A subset B
> > > A << B implies S\B << S\A
> > > A << B/\C iff A << B and A << C

> >
> > Is this the same as neighbourhood space defined as follows.
> >
> > (S, N), S a set, N a map S -> PPS (P for power set) and
> > i) x in S => N(x) =/= 0
> > ii) x in S, M in N(x) => x in M
> > iii) x in S, M in N(x) => (L superset M => L in N(x)
> > iv) x in S, L, M in N(x) => L intersect M in N(x)
> > v) x in S, M in N(x) => exists L in N(x) s.t.
> > L subset M and, forall y in L, L in N(y)
> > ?

>
> What I meant when asking are they the same was:
> a) can << be defined in terms of N, and can William's axioms be deduced as
> theorems from mine; and


LIkely not for the reasons I previously indicated.

> b) can N be defined in terms of <<, and can my axioms be deduced as theorems
> from William's?


Yes, I showed you how.

> And of course everyone knew that's what I meant.

Popular Presumption.

> I wrote of my axioms just for reasons of euphony, I think they are
> Hausdorff's. Again, someone more knowledgeable than I will know.


Here's from the web. Are these equivalent to yours?

The axioms formulated by Hausdorff (1919) for his concept of a
[32]topological space. These axioms describe the properties satisfied
by subsets of elements x in a [33]neighborhood [34]set E of x .

1. There corresponds to each point x at least one [35]neighborhood
U(x) , and each [36]neighborhood U(x) contains the point x .

2. If U(x) and V(x) are two [37]neighborhoods of the same point x ,
there must exist a [38]neighborhood W(x) that is a subset of both.

3. If the point y lies in U(x) , there must exist a [39]neighborhood
U(y) that is a [40]subset of U(x) .

4. For two different points x and y , there are two corresponding
[41]neighborhoods U(x) and U(y) with no points in common.

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