Talk:Closure (topology)
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I phrased the bit about density as a fact relating it to closure (expressed with "iff") rather than as a definition in terms of closure (expressed with "if"), since there are alternative definitions of dense sets. (And someday I'll probably edit the article on dense sets to mention them.) -- Toby Bartels 2002/05/08
A separate article is titled closed set. Should these two articles be merged? -- Michael Hardy, 2003 Aug 12
I don't think so; conceptually they are quite different, since one is a property of sets, while the other is an operation on them. To be sure, they are closely related and should be interlinked; but the concepts of open set, dense set, and the like are also closely related, yet separate. We can afford to have an article on each of these, eventually expanding them all like Open set is now. -- Toby Bartels 05:11, 25 Sep 2003 (UTC)
Alternative characterization
I have found this characterization of a closure useful. For any S in topological space X
1. Cl(S) is in topological space X
2. Cl(S) is closed
3. For any closed A in topological space X, A ⊇ Cl(S) iff A ⊇ S
The Interior can be defined similarly.
This seems to me a much less clumsy way of doing things than using the standard definition. Many results follow immediately by substituting various terms for A. Since CL(S) ⊇ CL(S) always holds, we also have CL(S) ⊇ S, CL(S) being closed by assumption. For closed S, we similarly have S ⊇ CL(S), hence S = CL(S). Jamie Oglethorpe 8 July 2004
- Yes, this seems to be another way of saying that Cl(S) is the smallest closed set containing S.