Connected space: Difference between revisions
imported>Richard Pinch (New entry, just a stub, with anchors) |
mNo edit summary |
||
| (5 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
{{subpages}} | |||
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. | 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. | ||
| Line 4: | Line 5: | ||
* The connected subsets of the [[real number]]s with the Euclidean metric topology are the [[interval]]s. | * The connected subsets of the [[real number]]s with the Euclidean metric topology are the [[interval]]s. | ||
* An [[indiscrete space]] is connected. | * An [[indiscrete space]] is connected. | ||
* A [[discrete space]] with more than one point is | * A [[discrete space]] with more than one point is not connected. | ||
==Properties== | |||
The image of a connected space under a [[continuous map]] is again connected. | |||
In conjunctions with the statement above, that the connected subsets of the [[real number]]s with the Euclidean metric topology are the [[interval]]s, this gives the [[Intermediate Value Theorem]]. | |||
==Connected component== | ==Connected component== | ||
| Line 17: | Line 23: | ||
* 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''. | ||
A path-connected space is connected, but not necessarily conversely. | |||
===Hyperconnected space=== | |||
A '''hyperconnected space''' or '''irreducible 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> (equivalently the space is not the union of proper closed subsets). | |||
A hyperconnected space is connected, but not necessarily conversely. Hyperconnectedness is open hereditary but not necessarily closed hereditary. Every topological space is [[homeomorphism|homeomorphic]] to a closed subspace of a hyperconnected space.<ref>{{cite journal | zbl=0813.54016 | author=Ajmal, N. | coauthors2=Kohli, J.K. | title=Properties of hyperconnected spaces, their mappings into Hausdorff spaces and embeddings into hyperconnected spaces | journal=Acta Math. Hung. | volume=60 | number=1-2 | pages=41-49 | year=1992 }}</ref> | |||
==References== | |||
{{reflist}}[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 11:00, 1 August 2024
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 not connected.
Properties
The image of a connected space under a continuous map is again connected.
In conjunctions with the statement above, that the connected subsets of the real numbers with the Euclidean metric topology are the intervals, this gives the Intermediate Value Theorem.
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)
A path-connected space is connected, but not necessarily conversely.
Hyperconnected space
A hyperconnected space or irreducible space is one in which the intersection of any two non-empty open sets is again non-empty[1] (equivalently the space is not the union of proper closed subsets).
A hyperconnected space is connected, but not necessarily conversely. Hyperconnectedness is open hereditary but not necessarily closed hereditary. Every topological space is homeomorphic to a closed subspace of a hyperconnected space.[2]