[Haskell-cafe] Category Theory woes

[Nick Rudnick]

Hi Hans,

agreed, but, in my eyes, you directly point to the problem:

* doesn't this just delegate the problem to the topic of limit 
operations, i.e., in how far is the term «closed» here more perspicuous?

* that's (for a very simple concept) the way that maths prescribes:
+ historical background: «I take "closed" as coming from being closed 
under limit operations - the origin from analysis.»
+ definition backtracking: «A closure operation c is defined by the 
property c(c(x)) = c(x). If one takes c(X) = the set of limit points of 
X, then it is the smallest closed set under this operation. The closed 
sets X are those that satisfy c(X) = X. Naming the complements of the 
closed sets open might have been introduced as an opposite of closed.»

418 bytes in my file system... how many in my brain...? Is it efficient, 
inevitable? The most fundamentalist justification I heard in this regard 
is: «It keeps people off from thinking the could go without the 
definition...» Meanwhile, we backtrack definition trees filling books, 
no, even more... In my eyes, this comes equal to claiming: «You have 
nothing to understand this beyond the provided authoritative definitions 
-- your understanding is done by strictly following these.»

Back to the case of open/closed, given we have an idea about sets -- we 
in most cases are able to derive the concept of two disjunct sets facing 
each other ourselves, don't we? The only lore missing is just a Bool: 
Which term fits which idea? With a reliable terminology using 
«bordered/unbordered», there is no ambiguity, and we can pass on 
reading, without any additional effort.

Picking such an opportunity thus may save a lot of time and even error 
-- allowing you to utilize your individual knowledge and experience. I 
have hope that this approach would be of great help in learning category 
theory.

All the best,

    Nick


Hans Aberg wrote:
> On 18 Feb 2010, at 14:48, Nick Rudnick wrote:
>
>> * 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),
>
> I take "closed" as coming from being closed under limit operations - 
> the origin from analysis. A closure operation c is defined by the 
> property c(c(x)) = c(x). If one takes c(X) = the set of limit points 
> of X, then it is the smallest closed set under this operation. The 
> closed sets X are those that satisfy c(X) = X. Naming the complements 
> of the closed sets open might have been introduced as an opposite of 
> closed.
>
>   Hans
>
>
>