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644:, M. Barge and J. Kennedy, in Open Problems in Topology, J. van Mill and G.M. Reed (Editors) Elsevier Science Publishers B.V. (North-Holland), 1990.
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are among the simplest examples of indecomposable homogeneous continua. They are neither arcwise connected nor locally connected.
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322:-axis. It is a one-dimensional continuum that is not arcwise connected, and it is locally disconnected at the points along the
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430:, is a one-dimensional planar Peano continuum that contains a homeomorphic image of any one-dimensional planar continuum.
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is a continuum that cannot be represented as the union of two proper subcontinua. A continuum
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402:-dimensional homogeneous continuum that is not contractible, and therefore different from an
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by an arc connecting (0,−1) and (1,sin(1)). It is a one-dimensional continuum whose
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287:. An arc is the simplest and most familiar type of a continuum. It is one-dimensional,
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There are two fundamental techniques for constructing continua, by means of
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is a subset of the plane that is the union of the graph of the function
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A nonempty compact connected metric space in point-set topology
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is a homogeneous hereditarily indecomposable planar continuum.
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A finite or countable product of continua is a continuum.
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A continuum that contains more than one point is called
376:. It is contractible and is the simplest example of an
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209:. A one-dimensional continuum is often called a
110:. A space homeomorphic to a subcontinuum of the
613:. Pure and Applied Mathematics, Marcel Dekker.
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398:+ 1)-dimensional Euclidean space. It is an
62:, or, less frequently, a compact connected
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642:Continuum Theory and Topological Dynamics
488:, then their intersection is a continuum.
390:is a space homeomorphic to the standard
469:} is a nested family of continua, i.e.
515:)} is an inverse sequence of continua
365:is a space homeomorphic to the closed
413:is an infinite-dimensional continuum.
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102:itself is a continuum is called a
74:devoted to the study of continua.
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632:Open problems in continuum theory
611:Continuum theory. An introduction
341:are all trivial, but it is not a
205:of a continuum usually means its
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333:is obtained by "closing up" the
314:≤ 1 with the segment −1 ≤
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637:Examples in continuum theory
192:hereditarily indecomposable
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929:Banach fixed-point theorem
593:Shape theory (mathematics)
428:Sierpinski universal curve
49:(plural: "continua") is a
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194:if every subcontinuum of
291:, and locally connected.
184:indecomposable continuum
132:if for every two points
30:Not to be confused with
380:-dimensional continuum.
335:topologist's sine curve
296:topologist's sine curve
243:is a homeomorphism and
175:is a continuum that is
984:Mathematics portal
884:Metrics and properties
870:Second-countable space
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275:; one also says that
207:topological dimension
32:Continuity (topology)
939:Invariance of domain
891:Euler characteristic
865:Bundle (mathematics)
448:nested intersections
426:, also known as the
949:Tychonoff's theorem
944:Poincaré conjecture
698:General (point-set)
609:Sam B. Nadler, Jr,
934:De Rham cohomology
855:Polyhedral complex
845:Simplicial complex
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343:contractible space
198:is indecomposable.
43:point-set topology
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838:fundamental group
526:coordinate spaces
424:Sierpinski carpet
289:arcwise connected
177:locally connected
70:is the branch of
16:(Redirected from
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60:metric space
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39:mathematical
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1004:Wikiversity
921:Key results
563:, then its
318:≤ 1 of the
227:is a space
130:homogeneous
78:Definitions
850:CW complex
791:Continuity
781:Closed set
740:cohomology
599:References
442:Properties
435:pseudo-arc
310:), 0 <
306:) = sin(1/
156:such that
98:such that
1029:geometric
1024:algebraic
875:Cobordism
811:Hausdorff
806:connected
723:Geometric
713:Continuum
703:Algebraic
417:Solenoids
269:endpoints
203:dimension
90:A subset
57:connected
47:continuum
41:field of
1056:Category
994:Wikibook
972:Category
860:Manifold
828:Homotopy
786:Interior
777:Open set
735:Homology
684:Topology
577:See also
394:in the (
392:n-sphere
218:Examples
72:topology
51:nonempty
1019:general
821:uniform
801:compact
752:Digital
604:Sources
388:-sphere
369:in the
231:to the
54:compact
37:In the
1014:Topics
816:metric
691:Fields
617:
406:-cell.
326:-axis.
255:(1) =
247:(0) =
796:Space
497:If {(
363:-cell
259:then
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235:. If
211:curve
615:ISBN
460:If {
450:and
433:The
422:The
409:The
367:ball
329:The
294:The
263:and
251:and
201:The
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136:and
45:, a
383:An
358:An
283:to
271:of
225:arc
223:An
190:is
182:An
140:in
128:is
106:of
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550:→
548:+1
540::
506:,
486:+1
478:⊇
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