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

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William Elliot

Posts: 1,494
Registered: 1/8/12
Re: Nhood Space
Posted: Jul 2, 2013 4:57 AM
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On Tue, 2 Jul 2013, Peter Percival wrote:

> > > > 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)
> > > > ?

> > > 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.

>
> Mine doesn't imply the separation axiom 4.


Without 4, are they equivalent? If so, then
how would you prove from 1-3, your axiom
iii) x in S, M in N(x) => (L superset M => L in N(x) ?




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