Talk:Compact element

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 Field:  Algebra

Way below[edit]

How can c be "way below" itself? --karlheg 08:35, 2004 Nov 24 (UTC)

Inconceivable, right? Perhaps the word does not mean what you think it means....
We say that an element x is called "way below y" if x is an element of every ideal I satisfying sup(I) ≥ y.
Since the principal ideal generated by y satisfies this condition, we know that any x which is way below y must be in it, i.e., must satisfy xy -- but not necessarily x<y.
(So the short answer is: the word "below" has to be interpreted as "below-or-equal".)
Aleph4 11:41, 14 June 2006 (UTC)
In view of this definition, the term "way below" merely duplicates in obscure language the preceding condition, which is the definition of being "way below" itself. I removed it. Zaslav 09:46, 22 March 2007 (UTC)

Introduction[edit]

The introduction is incomprehensible. Someone should write a clear and simple intro that does not require previous knowledge. I will see if I can do it, but others are welcome to the job. Zaslav 09:24, 22 March 2007 (UTC)

Standard[edit]

I removed the following remark: "Compact elements are standard." It makes no sense. Is "standard" a technical term? Then it needs explanation; please, someone, do that. If not, then what does the sentence mean? Zaslav 09:40, 22 March 2007 (UTC)

Examples[edit]

Given the complete lattice determined by the open sets of an arbitrary topological space ordered by set inclusion, the compact elements are not the compact sets of the space since in general compact sets are not open (and so would not even be members of this lattice). As far as I can tell, the name "compact" comes from the analogy with the topological definition, unless there is a restricted set of topological spaces for which this is true. Ron.garcia (talk) 20:37, 8 May 2010 (UTC)