Connected space: Difference between revisions
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* The [[rational number]]s as a [[subspace]] of the [[real number]]s with the Euclidean metric topology | * The [[rational number]]s as a [[subspace]] of the [[real number]]s with the Euclidean metric topology | ||
==Path-connected space== | ==Related concepts== | ||
===Path-connected space=== | |||
A '''path-connected space''' is one in which for any two points ''x'', ''y'' there exists a ''path'' from ''x'' to ''y'', that is, a [[continuous function]] <math>p: [0,1] \rightarrow X</math> such that ''p''(0)=''x'' and ''p''(1)=''y''. | A '''path-connected space''' is one in which for any two points ''x'', ''y'' there exists a ''path'' from ''x'' to ''y'', that is, a [[continuous function]] <math>p: [0,1] \rightarrow X</math> such that ''p''(0)=''x'' and ''p''(1)=''y''. | ||
===Hyperconnected space=== | |||
A '''hyperconnected space''' is one in which the intersection of any two non-empty open sets is again non-empty<ref>{{cite journal | id=Zbl 0664.54013 | author=Mathew, P.M. | title=On hyperconnected spaces | journal=Indian J. Pure Appl. Math. | volume=19 | number=12 | pages=1180-1184 | year=1988 | issn=0019-5588 }}</ref>. | |||
==References== | |||
{{reflist}} | |||
Revision as of 02:18, 27 December 2008
In topology, a connected space is a topological space in which there is no (non-trivial) subset which is simultaneously open and closed. Equivalently, the only continuous function from the space to a discrete space is constant. A disconnected space is one which is not connected.
Examples
- The connected subsets of the real numbers with the Euclidean metric topology are the intervals.
- An indiscrete space is connected.
- A discrete space with more than one point is nor connected.
Connected component
A connected component of a topological space is a maximal connected subset: that is, a subspace C such that C is connected but no superset of C is.
Totally disconnected space
A totally disconnected space is one in which the connected components are all singletons.
Examples
- A discrete space
- The Cantor set
- The rational numbers as a subspace of the real numbers with the Euclidean metric topology
Related concepts
Path-connected space
A path-connected space is one in which for any two points x, y there exists a path from x to y, that is, a continuous function such that p(0)=x and p(1)=y.
![{\displaystyle p:[0,1]\rightarrow X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e038832f49f4ea3480dcad8a4b14ecb3304d135a)
Hyperconnected space
A hyperconnected space is one in which the intersection of any two non-empty open sets is again non-empty[1].