NetNews Usenet Archive 1992 #26/NN_1992_26.iso/spool/sci/math/14606

Text File | 1992-11-08 | 3.0 KB | 62 lines

Newsgroups: sci.math
Path: sparky!uunet!comp.vuw.ac.nz!cc-server4.massey.ac.nz![e-mail]
From: [e-mail] (USENET News System)
Subject: Re: definition of topological space
Message-ID: <[e-mail]>
Organization: Massey University
References: <[e-mail]> <[e-mail]> <[e-mail]>
Date: Sun, 8 Nov 92 21:10:02 GMT
Lines: 51

In article <[e-mail]>, [e-mail] (G.J. McCaughan) writes:
>
> Here is another axiomatic definition of "topological space", which is
> probably more intuitive than the usual one in terms of open sets.
>
> A topological space is a set X of "points", together with (for each point)
> a class of subsets of X, called "neighbourhoods" of the point, such that:
>
> 1. If N is a nbhd of x then x is in N.
> 2. The intersection of two nbhds of x is a nbhd of x.
> 3. Anything containing a nbhd of x is a nbhd of x.
>
> This definition is equivalent to the one in terms of open sets. N is a nbhd
> of x iff it contains some open set containing x; on the other hand, U is open
> iff it is a nbhd of all its points. (This last is quite a good way of thinking
> about just what an open set is.)
>

Not quite, you need

4. Any nbhd N of x contains another nbhd M of x which is a nhbd of
each of its points.

This implies that there are open neighbourhoods.

It is probably the extra complication here which suggested working
with open sets. (It could also be that it seemed more elegant to have
an absolute condition - open, without regard to any specific point -
rather than a relative condition - a neighbourhood of a specific point).

One way of allowing one to work with neighbourhoods rather than
open sets while not complicating the axioms (until there is motivation
to do so) is to use the Frechet V-space notion.

A Frechet V-space is a set such that each point is equipped with a
neighbourhood system. There are no axioms, but definitions are carefully
phrased so that 1, 2 and 3 are unecessary. E.g. x is a point of accumulation
of a set A i, if, whenever N is a nbhd of x, then N-{x} contains
a point of A; while x is in the interior of A if there exists a nbhd N of x
such that N U {x} is contained in A. (If we take as an axiom that x is
in none of its nbhds, then we could use N instead of N-{x}, if x is in
all its nbhds, then we could use N instead of N U {x}).

It turns out that the axiom 4 is needed to avoid the following. With the
obvious definition of boundary, define the interior of a set to be the
set without the boundary, and the closure, to be the set together with
the boundary. The interior of the interior and the closure of the closure
are not necessarily the interior and closure unless axiom 4 holds.
(In fact Frechet defines the interior to be the largest open subset and the
closure to be the smallest closed subset). Without axiom 4, the interior and
closure (as I defined them, not as Frechet did) are not open (closed).