Nick Rudnick joerg.rudnick at t-online.de
Thu Feb 18 16:54:03 EST 2010

```Hi Mike,

of course... But in the same spirit, one could introduce a
straightforward extension, «partially bordered», which would be as least
as good as «clopen»... ;-)

I must admit we've come a little off the topic -- how to introduce to
category theory. The intent was to present some examples that
mathematical terminology culture is not that exemplary as one should
expect, but to motivate an open discussion about how one might «rename
refactor» category theory (of 2:48 PM).

I would be very interested in other people's proposals... :-)

Michael Matsko wrote:
>
> 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.rudnick at t-online.de>
> To: "Michael Matsko" <msmatsko at comcast.net>
> 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" <msmatsko at comcast.net>
>     To: "Nick Rudnick" <joerg.rudnick at 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.rudnick at t-online.de>
>     To: "Gregg Reynolds" <dev at mobileink.com>
>     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.rudnick at t-online.de <mailto:joerg.rudnick at 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
>             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
>
>
>
>
>     ------------------------------------------------------------------------
>
>     _______________________________________________