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I can't figure out the definition here. All I have is this, from the lead
what does that mean? If I take the unit interval, and then consider the right 2/3rds of it, and the left 2/3rds of it, then their union forms the whole interval. So is the unit interval decomposable?
If I take the buckethandle, and consider left 4/7ths of it, and the right 4/7ths of it, wouldn't the whole bucket-handle be the union of these two pieces? What am I missing? 67.198.37.16 ( talk) 17:37, 27 November 2017 (UTC)
The section History contains this sentence:
"In 1910 L. E. J. Brouwer described an indecomposable continuum which disproved a conjecture made by Arthur Moritz Schoenflies that the joint boundary of two open, connected, disjoint sets in was the union of two closed, connected proper subsets."
It is extremely easy to find two disjoint connected open sets in the plane, the intersection of whose boundaries is not the union of two closed connected proper subsets. (Imagine a thickened capital E next to its mirror image, so that their common boundary is the union of three disjoint closed intervals.)
I will hazard a guess that Schoenflies's (eventually disproven) conjecture was this:
"If the intersection of the boundaries of two disjoint open sets in the plane is a continuum, then it must be the union of two closed. connected proper subsets." 2601:200:C000:1A0:B4E6:9BA4:980:C6B4 ( talk) 20:09, 2 August 2021 (UTC)
"... an indecomposable continuum is a continuum that is indecomposable, i.e. ..."
This statement is a vacuous tautology. Instead the sentence should proceed directly to the material which follows the "i.e.", which is the actual definition:
"In point-set topology, an indecomposable continuum is a continuum that cannot be expressed as the union of any two of its proper subcontinua." 71.188.95.60 ( talk) 17:53, 18 January 2024 (UTC)
![]() | This article is rated Start-class on Wikipedia's
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I can't figure out the definition here. All I have is this, from the lead
what does that mean? If I take the unit interval, and then consider the right 2/3rds of it, and the left 2/3rds of it, then their union forms the whole interval. So is the unit interval decomposable?
If I take the buckethandle, and consider left 4/7ths of it, and the right 4/7ths of it, wouldn't the whole bucket-handle be the union of these two pieces? What am I missing? 67.198.37.16 ( talk) 17:37, 27 November 2017 (UTC)
The section History contains this sentence:
"In 1910 L. E. J. Brouwer described an indecomposable continuum which disproved a conjecture made by Arthur Moritz Schoenflies that the joint boundary of two open, connected, disjoint sets in was the union of two closed, connected proper subsets."
It is extremely easy to find two disjoint connected open sets in the plane, the intersection of whose boundaries is not the union of two closed connected proper subsets. (Imagine a thickened capital E next to its mirror image, so that their common boundary is the union of three disjoint closed intervals.)
I will hazard a guess that Schoenflies's (eventually disproven) conjecture was this:
"If the intersection of the boundaries of two disjoint open sets in the plane is a continuum, then it must be the union of two closed. connected proper subsets." 2601:200:C000:1A0:B4E6:9BA4:980:C6B4 ( talk) 20:09, 2 August 2021 (UTC)
"... an indecomposable continuum is a continuum that is indecomposable, i.e. ..."
This statement is a vacuous tautology. Instead the sentence should proceed directly to the material which follows the "i.e.", which is the actual definition:
"In point-set topology, an indecomposable continuum is a continuum that cannot be expressed as the union of any two of its proper subcontinua." 71.188.95.60 ( talk) 17:53, 18 January 2024 (UTC)