“Indecomposable continuum”的意思、由来-开放百科全书

In point-set topology, an indecomposable continuum is a continuum that is indecomposable, i.e. that cannot be expressed as the union of any two of its proper subcontinua. In 1910, L. E. J. Brouwer was the first to describe an indecomposable continuum.

Indecomposable continua have been used by topologists as a source of counterexamples. They also occur in dynamical systems.

Definitions

A continuum

is a nonempty compact connected metric space. The arc, the n-sphere, and the Hilbert cube are examples of path connected continua; the topologist's sine curve and Warsaw circle are examples of non-path connected continua. A subcontinuum

of a continuum

is a closed, connected subset of

. A space is nondegenerate if it is not equal to a single point. A continuum

is decomposable if there exist two subcontinua

and

such that

and

but

. A continuum which is not decomposable is an indecomposable continuum. A continuum

in which every subcontinuum is indecomposable is said to be hereditarily indecomposable. A composant of an indecomposable continuum

is a maximal set in which any two points lie within some proper subcontinuum of

. A continuum

is irreducible between and if

and no proper subcontinuum contains both points. An indecomposable continuum is irreducible between any two of its points.^[1]

History

Bucket handle example

Indecomposable continua are often constructed as the limit of a sequence of nested intersections, or (more generally) as the inverse limit of a sequence of continua. The buckethandle, or Brouwer–Janiszewski–Knaster continuum, is often used the simplest example of an indecomposable continuum, and can be so constructed (see upper right). Alternatively, take the Cantor ternary set

projected onto the interval

of the

-axis in the plane. Let

be the family of semicircles above the

-axis with center

and with endpoints on

(which is symmetric about this point). Let

be the family of semicircles below the

-axis with center the midpoint of the interval

and with endpoints in

. Let

be the family of semicircles below the

-axis with center the midpoint of the interval

and with endpoints in

. Then the union of all such

is the bucket handle.^[4]

The bucket handle admits no Borel transversal, that is there is no Borel set containing exactly one point from each composant.

Properties

In a sense, 'most' continua are indecomposable. Let

be an

-cell with metric

the set of all nonempty closed subsets of

, and

the hyperspace of all connected members of

equipped with the Hausdorff metric

defined by

. Then the set of nondegenerate indecomposable subcontinua of

is dense in

In dynamical systems

In 1932 George Birkhoff described his "remarkable closed curve", a homeomorphism of the annulus that contained an invariant continuum. Marie Charpentier showed that this continuum was indecomposable, the first link from indecomposable continua to dynamical systems. The invariant set of a certain Smale horseshoe map is the bucket handle. Marcy Barge and others have extensively studied indecomposable continua in dynamical systems.^[5]

See also

References

1. ^{{cite book|last1=Nadler|first1=Sam|title=Continuum Theory: An Introduction|date=2017|publisher=CRC Press|isbn=9781351990530|url=https://books.google.com/books?id=1R8uDwAAQBAJ|language=en}}
2. ^{{citation|journal=Mathematische Annalen|year=1910|volume= 68|issue =3|pages=422–434|title=Zur Analysis Situs|first=L. E. J.|last= Brouwer|doi=10.1007/BF01475781}}
3. ^{{cite book|last1=Cook|first1=Howard|last2=Ingram|first2=William T.|last3=Kuperberg|first3=Krystyna|last4=Lelek|first4=Andrew|last5=Minc|first5=Piotr|title=Continua: With the Houston Problem Book|date=1995|publisher=CRC Press|isbn=9780824796501|pages=103|url=https://books.google.com/books?id=bVcZGMOA4AEC&pg=PA103|language=en}}
4. ^{{cite book|last1=Ingram|first1=W. T.|last2=Mahavier|first2=William S.|title=Inverse Limits: From Continua to Chaos|date=2011|publisher=Springer Science & Business Media|isbn=9781461417972|pages=16|url=https://books.google.com/books?id=II3W2CZxDYMC&pg=PA16|language=en}}
5. ^{{cite journal|last1=Kennedy|first1=Judy|title=How Indecomposable Continua Arise in Dynamical Systems|journal=Annals of the New York Academy of Sciences|date=1 December 1993|volume=704|issue=1|pages=180–201|doi=10.1111/j.1749-6632.1993.tb52522.x|language=en|issn=1749-6632}}