Am Donnerstag 18 Februar 2010 19:19:36 schrieb 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?
It's fairly natural in German, abgeschlossen: closed, finished, complete; offen: open, ongoing.
* that's (for a very simple concept)
That concept (open and closed sets, topology more generally) is *not* very simple. It has many surprising aspects.
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).
Actually, that's incomplete, missing are - c(x) contains x - c(x) is minimal among the sets containing x with y = c(y).
If one takes c(X) = the set of limit points of
Not limit points, "Berührpunkte" (touching points).
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.»
But you can't understand it except by familiarising yourself with the definitions and investigating their consequences. The name of a concept can only help you remembering what the definition was. Choosing "obvious" names tends to be misleading, because there usually are things satisfying the definition which do not behave like the "obvious" name implies.
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.
And we'd be very wrong. There are sets which are simultaneously open and closed. It is bad enough with the terminology as is, throwing in the boundary (which is an even more difficult concept than open/closed) would only make things worse.
Picking such an opportunity thus may save a lot of time and even error -- allowing you to utilize your individual knowledge and experience. I
When learning a formal theory, individual knowledge and experience (except coming from similar enough disciplines) tend to be misleading more than helpful.
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
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