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633:, 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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311:-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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419:, 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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391:-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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276:. 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
365:. It is contractible and is the simplest example of an
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198:. A one-dimensional continuum is often called a
99:. A space homeomorphic to a subcontinuum of the
602:. Pure and Applied Mathematics, Marcel Dekker.
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387:+ 1)-dimensional Euclidean space. It is an
51:, or, less frequently, a compact connected
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631:Continuum Theory and Topological Dynamics
477:, then their intersection is a continuum.
379:is a space homeomorphic to the standard
458:} is a nested family of continua, i.e.
504:)} is an inverse sequence of continua
354:is a space homeomorphic to the closed
402:is an infinite-dimensional continuum.
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91:itself is a continuum is called a
63:devoted to the study of continua.
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621:Open problems in continuum theory
600:Continuum theory. An introduction
330:are all trivial, but it is not a
194:of a continuum usually means its
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322:is obtained by "closing up" the
303:≤ 1 with the segment −1 ≤
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626:Examples in continuum theory
181:hereditarily indecomposable
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918:Banach fixed-point theorem
582:Shape theory (mathematics)
417:Sierpinski universal curve
38:(plural: "continua") is a
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183:if every subcontinuum of
280:, and locally connected.
173:indecomposable continuum
121:if for every two points
19:Not to be confused with
369:-dimensional continuum.
324:topologist's sine curve
285:topologist's sine curve
232:is a homeomorphism and
164:is a continuum that is
973:Mathematics portal
873:Metrics and properties
859:Second-countable space
343:
341:
264:; one also says that
196:topological dimension
21:Continuity (topology)
928:Invariance of domain
880:Euler characteristic
854:Bundle (mathematics)
437:nested intersections
415:, also known as the
938:Tychonoff's theorem
933:Poincaré conjecture
687:General (point-set)
598:Sam B. Nadler, Jr,
923:De Rham cohomology
844:Polyhedral complex
834:Simplicial complex
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332:contractible space
187:is indecomposable.
32:point-set topology
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827:fundamental group
515:coordinate spaces
413:Sierpinski carpet
278:arcwise connected
166:locally connected
59:is the branch of
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1051:Continuum theory
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57:Continuum theory
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360:Euclidean space
328:homotopy groups
268:is an arc from
256:are called the
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162:Peano continuum
101:Euclidean plane
83:of a continuum
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759:Key concepts
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28:mathematical
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993:Wikiversity
910:Key results
552:, then its
307:≤ 1 of the
216:is a space
119:homogeneous
67:Definitions
839:CW complex
780:Continuity
770:Closed set
729:cohomology
588:References
431:Properties
424:pseudo-arc
299:), 0 <
295:) = sin(1/
145:such that
87:such that
1018:geometric
1013:algebraic
864:Cobordism
800:Hausdorff
795:connected
712:Geometric
702:Continuum
692:Algebraic
406:Solenoids
258:endpoints
192:dimension
79:A subset
46:connected
36:continuum
30:field of
1045:Category
983:Wikibook
961:Category
849:Manifold
817:Homotopy
775:Interior
766:Open set
724:Homology
673:Topology
566:See also
383:in the (
381:n-sphere
207:Examples
61:topology
40:nonempty
1008:general
810:uniform
790:compact
741:Digital
593:Sources
377:-sphere
358:in the
220:to the
43:compact
26:In the
1003:Topics
805:metric
680:Fields
606:
395:-cell.
315:-axis.
244:(1) =
236:(0) =
785:Space
486:If {(
352:-cell
248:then
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224:. If
200:curve
604:ISBN
449:If {
439:and
422:The
411:The
398:The
356:ball
318:The
283:The
252:and
240:and
190:The
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125:and
34:, a
372:An
347:An
272:to
260:of
214:arc
212:An
179:is
171:An
129:in
117:is
95:of
1047::
539:→
537:+1
529::
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475:+1
467:⊇
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160:A
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