Timeline for Are there any important mathematical concepts without discrete analog?
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 20, 2012 at 2:28 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Aug 20, 2012 at 2:01 | comment | added | Benjamin Steinberg | I strongly would argue finiteness is the analogue of compactness for discrete. Just compare the representation theory of compact groups and finite groups. | |
| Mar 30, 2010 at 19:33 | comment | added | Tomaž Pisanski | I am coming from graph theory. I was fascinated by the fact, that "conectedness" and "path conectedness" coincide for graphs. Obviously, one has to prove that. However, the topological notion of a "path" corresponds to the idea of a "trail" in graph theory, while an "open set" corresponds to a union of connected components of graphs. | |
| Mar 9, 2010 at 8:52 | comment | added | Yemon Choi | It should also be pointed out that in some sense discreteness and compactness sit at opposite ends of the spectrum of locally compact spaces, so that it's not clear what a "discrete analogue" (as opposed to a "quantitative, finite analogue" might be | |
| Mar 9, 2010 at 8:50 | comment | added | Yemon Choi | I don't know if compactness is a "generalization of finiteness properties" as such, but it certainly gets used as a substitute for finiteness all the time. There are various parts of Banach space/Banach algebra theory where a desire to interchange the order of various iterated limits can be done by judicious appeal to weak compactness of various sets. | |
| Mar 9, 2010 at 0:08 | comment | added | Douglas Zare | The interior of a subset of a simplicial complex makes sense, so I've struck that out. It's not an inherently topological concept. I wouldn't consider $\mathbb Q^n$ discrete. I'm not convinced that compactness or boundedness are attempts to generalize important properties of finite sets rather than of $[0,1]$. Thanks for bringing up that blog entry. I certainly expected that my answer would bring out contrary ideas, and I hope people will contribute more. | |
| Mar 8, 2010 at 23:57 | history | edited | Douglas Zare | CC BY-SA 2.5 |
Added strikeout for the interior of a set.
|
| Mar 8, 2010 at 23:51 | comment | added | gowers | It amuses me that this answer has been accepted when discrete mathematicians would use analogues of every single one of these. I recommend a look at this post of Terry Tao: terrytao.wordpress.com/2007/05/23/… | |
| Mar 8, 2010 at 22:46 | comment | added | darij grinberg | Actually, boundedness is one of the possible continuous generalization of finiteness. Compactness is another. As for "the interior of a set", there is linear optimization where one is talking about interiors of polytopes, and while it is usually done in R^n, it could equally well be studied over Q^n or (any ordered field)^n, and actually is a combinatorial science where discrete algorithms such as the simplex method matter, and the continuous structure of the field is just a red herring. | |
| Mar 8, 2010 at 22:41 | comment | added | Dan Piponi | I'd agree that some of these discrete analogues can be vacuous, but isn't that the point? For example, when we study compact sets in topology are we not, at least sometimes, trying to find non-trivial analogues of results that are trivially true of finite sets in the discrete case? | |
| Mar 8, 2010 at 22:24 | comment | added | Mariano Suárez-Álvarez | It depends on what you mean by obvious... One example: the interior of a set $X$ of vertices in a graph may very well be defined as that subset of those elements in $X$ all of whose neighbors are in $X$. | |
| Mar 8, 2010 at 22:20 | vote | accept | Tomaž Pisanski | ||
| Mar 8, 2010 at 22:13 | history | answered | Douglas Zare | CC BY-SA 2.5 |