File spoon-archives/heidegger.archive/heidegger_2003/heidegger.0306, message 177


Date: Mon, 30 Jun 2003 16:25:27 +0200
From: Michael Hoppe <mh-AT-michael-hoppe.de>
Subject: Re: The dimension of a manifold


>   > The delta-epsilon business is a little misleading as it assumes, in a
>>  sense, a metric space. The topological notion (using, for example, a
>>  definition from Hall) is as follows: Let S and T be spaces
>>  [topological] and f:S->T a mapping. The f is said to be continuous at
>>  a point s [if one wants this can be interpreted as an element] of S
>>  if and only if, given any open subset G of T such that if s is a
>>  point of the inverse image under f of G [sorry, there is no nice way
>>  of doing this in email], there exists an open subset of V such that s
>>  is a point of V and V is contained in the inverse image under f of G
>>  [ well, a possible way might be f^-1(G)]
>
>  Do you mean "an open subset V" rather than "an open subset of V"?

And the given definition is a bit clumsy.  A mapping f: S -> T 
between the topological spaces S and T is continuous in the point s 
in S, iff the inverse of every neighbourhood of f(s) in T under f is 
open in S, in short: inverse images of open sets are open.

>  My memory is a bit hazy here. Is an open subset of a topological 
> space a subset in
>  which every element has a neighbourhood within the subset?

Given a set M, a topology T of M is a set of subsets of M with the 
properties that for any family of sets of T their union is contained 
in T and the section of any two sets in T is contained in T.  The 
elements of T are called "the open sets of M".

>   And neighbourhood in a
>  topological space is an axiomatic notion of points close to a given point?

Not quite.  A neighbourhood of a point p of the topological space M 
is an open set (i.e., an element of T) containing p.  And even more. 
As there are many different topoligies of a given set M, e.g., the 
reell number line, the "dictance" between two points or 
"connectedness" of a subset of M depends solely on the choosed 
topology: there exist topologies if the reell number line in which 
the interval [0, 1] is disconnected.

>  What you have provided is a definitional understanding of a function f being
>  continuous at a given point s, but isn't the notion of 
> neighbourhood the site of
>  the more originary notion of continuity, of something 'hanging 
> together'? Could it
>  be said that with the topological notion of neighbourhood the 
> ontological problem
>  of closeness and continuity and connectednes is somehow swept under 
> the carpet?

Yes, indeed.  Mathematicians talk about their subjects, and they do 
not invent new words for the mathematical things but use common words 
instead.  And those formerly commonly used words loose their original 
meaning: their new meaning is always only given in their definition.

     And a "closed" subset of a topological space is (per 
definitionem) nothing else than the complement of an open set of M.

>   > Also there is perhaps an interesting definition for connected: A
>>  subset A of a space S is said to be connected if and only if there
>>  exists no continuous mapping f:A->R [R here stands for the reals, but
>>  think of it as an index set] such that f(A) consists of exactly two
>>  points.

Even more "abstract": A topological set M is said to be connected, 
iff only M and the empty set (which is contained in each topology) 
are as well open and closed.

>  This interestingly inverts Aristotle's sequence.

It has to.

>  Intuitively (i.e. for a sensous looking-at), this is obvious (i.e. 
> unconcealed).

But if the mathematical things show themselves via sensous looking-at 
-- and they do -- this Anschauung as Anschauung might be misleading. 
There are e.g., metric spaces in which spheres have weird properties. 
Let S(p, r) denote the sphere aroung p with radius r, that is the set 
of all points, whose distance from p is less than r.  Then, if q is 
in S(x, r), it follows that S(p, r) = S(q, r).  Or, if two spheres 
have one point in common, one of the spheres is contained in the 
other.  Even weirder: the distance of two open spheres of radius r, 
which are contained in a closed sphere of radius r, is r ...

Michael
-- 
-= Michael Hoppe <www.michael-hoppe.de>, <mh-AT-michael-hoppe.de> =-----
-= Key fingerprint = 74 FD 0A E3 8B 2A 79 82 25 D0 AD 2B 75 6A AE 63
-= PGP public key ID 0xE0A5731D  =-----------------------------------


     --- from list heidegger-AT-lists.village.virginia.edu ---

   

Driftline Main Page

 

Display software: ArchTracker © Malgosia Askanas, 2000-2005