Nick,
That is correct. An open set contains no point on its boundary.
A closed set contains its boundary, i.e. for a closed set c,
Closure(c) = c.
Note that for a general set, which is neither closed or open (say
the half closed interval (0,1]), may contain points on its boundary.
Every set contains its interior, which is the part of the set without
its boundary and is contained in its closure - for a given set x,
Interior(x) is a subset of x is a subset of Closure(x).
Mike
----- Original Message -----
From: "Nick Rudnick" <joerg.rudn...@t-online.de>
To: "Michael Matsko" <msmat...@comcast.net>
Cc: haskell-cafe@haskell.org
Sent: Thursday, February 18, 2010 3:15:49 PM GMT -05:00 US/Canada Eastern
Subject: Re: Fwd: [Haskell-cafe] Category Theory woes
Hi Mike,
so an open set does not contain elements constituting a
border/boundary of it, does it?
But a closed set does, doesn't it?
Cheers,
Nick
Michael Matsko wrote:
----- Forwarded Message -----
From: "Michael Matsko" <msmat...@comcast.net>
To: "Nick Rudnick" <joerg.rudn...@t-online.de>
Sent: Thursday, February 18, 2010 2:16:18 PM GMT -05:00 US/Canada
Eastern
Subject: Re: [Haskell-cafe] Category Theory woes
Gregg,
Topologically speaking, the border of an open set is called the
boundary of the set. The boundary is defined as the closure of
the set minus the set itself. As an example consider the open
interval (0,1) on the real line. The closure of the set is [0,1],
the closed interval on 0, 1. The boundary would be the points 0
and 1.
Mike Matsko
----- Original Message -----
From: "Nick Rudnick" <joerg.rudn...@t-online.de>
To: "Gregg Reynolds" <d...@mobileink.com>
Cc: "Haskell Café List" <haskell-cafe@haskell.org>
Sent: Thursday, February 18, 2010 1:55:31 PM GMT -05:00 US/Canada
Eastern
Subject: Re: [Haskell-cafe] Category Theory woes
Gregg Reynolds wrote:
On Thu, Feb 18, 2010 at 7:48 AM, Nick Rudnick
<joerg.rudn...@t-online.de <mailto:joerg.rudn...@t-online.de>>
wrote:
IM(H??)O, a really introductive book on category theory
still is to be written -- if category theory is really
that fundamental (what I believe, due to its lifting of
restrictions usually implicit at 'orthodox maths'), than
it should find a reflection in our every day's common
sense, shouldn't it?
Goldblatt works for me.
Accidentially, I have Goldblatt here, although I didn't read it
before -- you agree with me it's far away from every day's common
sense, even for a hobby coder?? I mean, this is not «Head first
categories», is it? ;-)) With «every day's common sense» I did not
mean «a mathematician's every day's common sense», but that of,
e.g., a housewife or a child...
But I have became curious now for Goldblatt...
* the definition of open/closed sets in topology with the
boundary elements of a closed set to considerable extent
regardable as facing to an «outside» (so that reversing
these terms could even appear more intuitive, or
«bordered» instead of closed and «unbordered» instead of
open),
Both have a border, just in different places.
Which elements form the border of an open set??
As an example, let's play a little:
Arrows: Arrows are more fundamental than objects, in fact,
categories may be defined with arrows only. Although I
like the term arrow (more than 'morphism'), I intuitively
would find the term «reference» less contradictive with
the actual intention, as this term
Arrows don't refer.
A *referrer* (object) refers to a *referee* (object) by a
*reference* (arrow).
Categories: In every day's language, a category is a
completely different thing, without the least
Not necesssarily (for Kantians, Aristoteleans?)
Are you sure...?? See
http://en.wikipedia.org/wiki/Categories_(Aristotle) ...
If memory serves, MacLane says somewhere that he and
Eilenberg picked the term "category" as an explicit play on
the same term in philosophy.
In general I find mathematical terminology well-chosen and
revealing, if one takes the trouble to do a little digging.
If you want to know what terminological chaos really looks
like try linguistics.
;-) For linguistics, granted... In regard of «a little digging»,
don't you think terminology work takes a great share, especially
at interdisciplinary efforts? Wouldn't it be great to be able to
drop, say 20% or even more, of such efforts and be able to
progress more fluidly ?
-g
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