Talk:Lebesgue covering dimension

message: The definition given uses the term "refinement", but incorrectly. A refinement is not a subcover, the elements of the refinement need only be open subsets of the original cover. I don't know if the definition of LCD is correct with "refinement" or "subcover", though. 4.31.92.84 05:15, 17 Dec 2004 (UTC)
replies: text: As one of the authors, I'll comment that the page probably needs checking anyway. It is partly based on what I recall of the proof of the simplicial approximation theorem. Charles Matthews 14:10, 17 Dec 2004 (UTC)

text: Never mind. The key to lebesgue defnition is that every (finite?) open cover must have a (finite?) refinement where each point is contained in no more than n+1 sets of the refinement. The disk is 2D, because it has open covers that cannot be refined so that each point is only in 2 sets; 3 is the minimum. The Menger sponge still seems to be 1D. I don't know if the finite/infinite distinction might change this. linas 15:17, 19 May 2005 (UTC)
replies: text: I think should be every open cover (not necesarily finite) with a locally finite refinement... If there exist open covers without locally finite refinements the top. dim. would be infinite.
Hmm. The edit by User:Kuratowski's Ghost seems really really wrong to me. The original wording, refinment was correct, subcover is wrong. Here's a counter-example: Take the unit disk in 2D and slice it up into 12 pie slices. Grow each pie slice a bit, so each pie slice overlaps. That's an open covering of the unit disk by 12 open sets. Points near the center of the disk now belong to 12 open sets, and its impossible to discard any of the twelve and still have a cover. This implies that the 2D disk is 11-dimensional, clearly false. So that definition is broken.  By contrast, those 12 slices can be refined so that the points in the center are covered by no more than three open sets. So the earlier version of the article was correct. linas 03:29, 20 May 2005 (UTC)
thread_title: Refinement vs. subcover

message: :I'm planning on adding the following text to the main article real soon now.  Please review.  
The definition of the Lebesgue covering dimension can be used to build some unusual toplogical sets, such as the Sierpinski carpet.  A construction can proceed as follows. 
Consider, for example, a finite open covering for the two-dimensional unit disk. This covering can always be refined so that no point in the disk belongs to more than three sets.  Fixing this covering, remove all of the points in the disk that belong to three sets.  Depending on the refinement, this will leave possibly one or more holes in the disk.  The remaining object is again two-dimensional, and again has a finite open cover. The process of selecting a cover and refining, and then punching out holes can be repeated, ad infinitum. The resulting object is homeomorphic to the Sierpinski carpet. What is curious about this construction is that the carpet has a Lebesgue covering dimension of one, and not two. Given any open covering of the carpet, one can always find a refinement such that every point belongs to at most two sets. The proof of this is essentially by contradiction: were there a covering which required membership to three sets, then the affected area would have been punched out during the construction phase.  As open covers are at most countable, every such case is handled during construction. Similar constructions can be performed in higher dimensions; the three-dimensional analogue is called the Menger sponge. Curiously, the Lebesgue covering dimension of the Menger sponge is again one.  
This construction of the Sierpinski carpet is at best incomplete because it assumes that only a countable number of different covers are possible and that every cover of the sponge extends to a cover of some step in the construction. Neither of these facts are evident. Grubb257 17:57, 27 September 2007 (UTC)
thread_title: Construction of the Sierpinski Carpet

message: An earlier version of the article spoke of a refinements instead of subcovers, but the definition of refinement that was given was that of a subcover, so I changed it to subcover. But this indeed doesn't make sense as the examples show, I assume it really meant refinement but explained it incorrectly, will change again. Kuratowski's Ghost 09:40, 20 May 2005 (UTC)
thread_title:  Refinement or Subcover? 

message: Am I right in thinking that "any object of Lebesgue dimension one", means  compact metrizable spaces of Lesbesgue dimension one. Certainly object can't mean topological space or we could take a discrete space with too big a cardinality as a counterexample. I'm not CERTAIN we need metrizable (what is the dimension of {0,1}^(big cardinal)?) I'd actually really like to know. A Geek Tragedy 14:59, 11 February 2007 (UTC)
thread_title: Menger sponge universal property

message: Is the topologic dimension the same as the spectral dimension?  I have been unable to locate an article that defines spectral dimension.  Could someone knowledgeable about this subject please add the definition of "spectral dimension" to wikipedia?  Thank you.  —Preceding unsigned comment added by 129.186.148.15 (talk) 18:29, 23 August 2010 (UTC)
thread_title:  Spectral Dimension 

message: Hi, I'm copying here the comment I received about the illustrations. --Sigmundur (talk) 08:02, 23 November 2010 (UTC)
replies: text:  Not quite sure what you mean, the picture may also be unclear. The idea is that there are four overlapping sets; the path thus also crosses the red set, even though it never crosses a point that is exclusively red. The refinements are then made so that every point on the line belongs to at most two sets in the refinement. Please comment further if I didn't address the problem. --Sigmundur (talk) 08:02, 23 November 2010 (UTC)
replies: text: FWIW, I also noted this issue and added the dubious tag.  70.247.167.104 (talk) 03:26, 6 January 2014 (UTC)
thread_title:  Illustrations 

message: Being an interested lay person, I still chuckle reading that header in such topics - the subject matter IS technical !  OK, so how would / could you at least introduce the topic in lay terms ? Is that even possible considering the need for the underlying "technical" terminology, e.g. dimensionality, open cover, etc. ?  The images are very helpful, but for me at least, still don't lead to that eureka moment.  Another thing that might help is a section on how this fits into the abstract notion of "spaces" as this term for lay people usually connotes Euclidean, physical space. Perhaps a text wrapper around Mathematical Spaces also with reference to sets and their basis for the general notion of topological spaces  - thanks !  — Preceding unsigned comment added by 98.71.193.159 (talk) 19:04, 24 July 2013 (UTC)
replies: text: I just wrote a non-technical informal discussion section that tries to explain the idea without the technicalities. 67.198.37.16 (talk) 22:33, 3 October 2020 (UTC)
thread_title:  "This article may be too technical for most readers to understand." 

message: The section Properties has a passage about "(Ostrand's theorem on colored dimension.)"
But I can find no reference to anything called a "colored dimension".
Is this just unbelievably sloppy editing? 2601:200:C000:1A0:80B1:F028:8EEF:3BF8 (talk) 20:27, 4 November 2021 (UTC)
thread_title:  Is this a mistake?