Connected space: Difference between revisions

Revision as of 17:13, 26 January 2012

This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spaces

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This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

This article is about a basic definition in topology.
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Definition

Symbol-free definition

A topological space is said to be connected if it satisfies the following equivalent conditions:

It cannot be expressed as a disjoint union of two nonempty open subsets
It cannot be expressed as a disjoint union of two nonempty closed subsets
It has no clopen subsets other than the empty subspace and the whole space

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Path-connected space	path joining any two points	path-connected implies connected	connected not implies path-connected	\|FULL LIST, MORE INFO
Simply connected space	path-connected, trivial fundamental group			\|FULL LIST, MORE INFO
Contractible space	homotopy-equivalent to a point			\|FULL LIST, MORE INFO

Opposite properties

Totally disconnected space

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
product-closed property of topological spaces	Yes	connectedness is product-closed	Suppose $X_{i}, i \in I$ , are all connected spaces. Then, the Cartesian product $\prod_{i \in I} X_{i}$ is also a connected space with the product topology.
box product-closed property of topological spaces	No	connectedness is not box product-closed	It is possible to have $X_{i}, i \in I$ all connected spaces such that the Cartesian product $\prod_{i \in I} X_{i}$ is not connected in the box topology.
coarsening-preserved property of topological spaces	Yes	connectedness is coarsening-preserved	If $X$ is connected under a topology $τ$ , it remains connected when we pass to a coarser topology than $τ$ .
continuous image-closed property of topological spaces	Yes	connectedness is continuous image-closed	If $X$ is a connected space and $Y$ is the image of $X$ under a continuous map, then $Y$ is also connected.

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 148 (formal definition)
Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 11 (formal definition)

@@ Line 35: / Line 35: @@
 ==Metaproperties==
-{{DP-closed}}
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
-Any product of connected spaces is connected in the [[product topology]] (this is true for finite, as well as infinite, products). {{proofat|[[Connectedness is product-closed]]}}
+|-
+| [[satisfies metaproperty::product-closed property of topological spaces]]|| Yes || [[connectedness is product-closed]] || Suppose <math>X_i, i \in I</math>, are all connected spaces. Then, the Cartesian product <math>\prod_{i \in I} X_i</math> is also a connected space with the [[product topology]].
+|-
-{{box-product-closed}}
+| [[dissatisfies metaproperty::box product-closed property of topological spaces]] || No || [[connectedness is not box product-closed]] || It is possible to have <math>X_i, i \in I</math> all connected spaces such that the Cartesian product <math>\prod_{i \in I} X_i</math> is ''not'' connected in the [[box topology]].
+|-
-A box product of connected spaces is connected.
+| [[satisfies metaproperty::coarsening-preserved property of topological spaces]] || Yes || [[connectedness is coarsening-preserved]] || If <math>X</math> is connected under a topology <math>\tau</math>, it remains connected when we pass to a [[coarser topology]] than <math>\tau</math>.
+|-
-{{coarsening-preserved}}
+| [[satisfies metaproperty::continuous image-closed property of topological spaces]] || Yes || [[connectedness is continuous image-closed]] || If <math>X</math> is a connected space and <math>Y</math> is the image of <math>X</math> under a continuous map, then <math>Y</math> is also connected.
+|}
-Switching to a coarser topology continues to keep the topological space connected. This is because if there is a proper nonempty clopen subset in the coarser topology, it would also be there in the finer topology.
-{{connected union-closed}}
-If a [[topological space]] is the union of a family of subsets, with the property that they all have a nonempty intersection, then the whole space is connected.
-A slight variant is this: if a topological space is the union of a subspace and a family of other subspaces that each intersect this subspace nontrivially, and all these subspaces are connected, then the whole space is connected.
-{{continuous image-closed}}
-If <math>X</math> is a connected space and <math>Y</math> is the image of <math>X</math> via a [[continuous map]], then <math>Y</math> is also connected. {{proofat|[[connectedness is continuous image-closed]]}}
 ==References==