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Hi Mike,<br>
<br>
<br>
of course... But in the same spirit, one could introduce a
straightforward extension, «partially bordered», which would be as
least as good as «clopen»... ;-)<br>
<br>
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). <br>
<br>
I would be very interested in other people's proposals... :-)<br>
<br>
Michael Matsko wrote:
<blockquote
cite="mid:813798396.4030621266527809066.JavaMail.root@sz0009a.westchester.pa.mail.comcast.net"
type="cite">
<style type="text/css">p { margin: 0; }</style>
<div style="font-family: Arial; font-size: 12pt; color: rgb(0, 0, 0);">
<p>Nick,</p>
<p> </p>
<p> That is correct. An open set contains no point on its
boundary. </p>
<p> </p>
<p> A closed set contains its boundary, i.e. for a closed set c,
Closure(c) = c. </p>
<p> </p>
<p> 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). </p>
<p> </p>
<p>Mike</p>
<p> <br>
----- Original Message -----<br>
From: "Nick Rudnick" <a class="moz-txt-link-rfc2396E" href="mailto:joerg.rudnick@t-online.de"><joerg.rudnick@t-online.de></a><br>
To: "Michael Matsko" <a class="moz-txt-link-rfc2396E" href="mailto:msmatsko@comcast.net"><msmatsko@comcast.net></a><br>
Cc: <a class="moz-txt-link-abbreviated" href="mailto:haskell-cafe@haskell.org">haskell-cafe@haskell.org</a><br>
Sent: Thursday, February 18, 2010 3:15:49 PM GMT -05:00 US/Canada
Eastern<br>
Subject: Re: Fwd: [Haskell-cafe] Category Theory woes<br>
<br>
Hi Mike,<br>
<br>
so an open set does not contain elements constituting a border/boundary
of it, does it?<br>
<br>
But a closed set does, doesn't it?<br>
<br>
Cheers,<br>
<br>
Nick<br>
<br>
Michael Matsko wrote: </p>
<blockquote
cite="mid:1878594958.3984921266520604735.JavaMail.root@sz0009a.westchester.pa.mail.comcast.net">
<style>p { margin: 0; }</style>
<div
style="font-size: 12pt; color: rgb(0, 0, 0); font-family: Arial;"><br>
----- Forwarded Message -----<br>
From: "Michael Matsko" <a moz-do-not-send="true"
class="moz-txt-link-rfc2396E" href="mailto:msmatsko@comcast.net"
target="_blank"><msmatsko@comcast.net></a><br>
To: "Nick Rudnick" <a moz-do-not-send="true"
class="moz-txt-link-rfc2396E" href="mailto:joerg.rudnick@t-online.de"
target="_blank"><joerg.rudnick@t-online.de></a><br>
Sent: Thursday, February 18, 2010 2:16:18 PM GMT -05:00 US/Canada
Eastern<br>
Subject: Re: [Haskell-cafe] Category Theory woes<br>
<br>
<style>p { margin: 0; }</style>
<div
style="font-size: 12pt; color: rgb(0, 0, 0); font-family: Arial;">
<p><font face="Times New Roman">Gregg,</font></p>
<p> </p>
<p><font face="Times New Roman"> 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.</font></p>
<p> </p>
<p><font face="Times New Roman">Mike Matsko</font></p>
<p><br>
----- Original Message -----<br>
From: "Nick Rudnick" <a moz-do-not-send="true"
class="moz-txt-link-rfc2396E" href="mailto:joerg.rudnick@t-online.de"
target="_blank"><joerg.rudnick@t-online.de></a><br>
To: "Gregg Reynolds" <a moz-do-not-send="true"
class="moz-txt-link-rfc2396E" href="mailto:dev@mobileink.com"
target="_blank"><dev@mobileink.com></a><br>
Cc: "Haskell Café List" <a moz-do-not-send="true"
class="moz-txt-link-rfc2396E" href="mailto:haskell-cafe@haskell.org"
target="_blank"><haskell-cafe@haskell.org></a><br>
Sent: Thursday, February 18, 2010 1:55:31 PM GMT -05:00 US/Canada
Eastern<br>
Subject: Re: [Haskell-cafe] Category Theory woes<br>
<br>
Gregg Reynolds wrote: </p>
<blockquote
cite="mid:75cc17ac1002180924r6d8eb5f4r3a7ddebd9825c69f@mail.gmail.com">
<div class="gmail_quote">On Thu, Feb 18, 2010 at 7:48 AM, Nick
Rudnick <span dir="ltr"><<a moz-do-not-send="true"
href="mailto:joerg.rudnick@t-online.de" target="_blank">joerg.rudnick@t-online.de</a>></span>
wrote:<br>
<blockquote class="gmail_quote"
style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<div>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?<br>
<br>
</div>
</blockquote>
<div><br>
Goldblatt works for me.<br>
</div>
</div>
</blockquote>
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...
<br>
<br>
But I have became curious now for Goldblatt...<br>
<blockquote
cite="mid:75cc17ac1002180924r6d8eb5f4r3a7ddebd9825c69f@mail.gmail.com">
<div class="gmail_quote">
<div> </div>
<blockquote class="gmail_quote"
style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<div><br>
* 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), </div>
</blockquote>
<div><br>
Both have a border, just in different places.<br>
</div>
</div>
</blockquote>
Which elements form the border of an open set??<br>
<blockquote
cite="mid:75cc17ac1002180924r6d8eb5f4r3a7ddebd9825c69f@mail.gmail.com">
<div class="gmail_quote">
<div><br>
</div>
<blockquote class="gmail_quote"
style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<div><br>
As an example, let's play a little:<br>
<br>
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<br>
<br>
</div>
</blockquote>
<div>Arrows don't refer. <br>
</div>
</div>
</blockquote>
A *referrer* (object) refers to a *referee* (object) by a *reference*
(arrow). <br>
<blockquote
cite="mid:75cc17ac1002180924r6d8eb5f4r3a7ddebd9825c69f@mail.gmail.com">
<div class="gmail_quote">
<div> </div>
<blockquote class="gmail_quote"
style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
<div>Categories: In every day's language, a category is a
completely different thing, without the least </div>
</blockquote>
<div><br>
Not necesssarily (for Kantians, Aristoteleans?)</div>
</div>
</blockquote>
Are you sure...?? See <a moz-do-not-send="true"
class="moz-txt-link-freetext"
href="http://en.wikipedia.org/wiki/Categories_%28Aristotle%29"
target="_blank">http://en.wikipedia.org/wiki/Categories_(Aristotle)</a>
...<br>
<blockquote
cite="mid:75cc17ac1002180924r6d8eb5f4r3a7ddebd9825c69f@mail.gmail.com">
<div class="gmail_quote">
<div> If memory serves, MacLane says somewhere that he and
Eilenberg picked the term "category" as an explicit play on the same
term in philosophy.<br>
</div>
</div>
</blockquote>
<blockquote
cite="mid:75cc17ac1002180924r6d8eb5f4r3a7ddebd9825c69f@mail.gmail.com">
<div class="gmail_quote">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.<br>
</div>
</blockquote>
;-) 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 ?<br>
<blockquote
cite="mid:75cc17ac1002180924r6d8eb5f4r3a7ddebd9825c69f@mail.gmail.com">
<div class="gmail_quote"><br>
-g<br>
</div>
<br>
</blockquote>
<br>
<br>
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