Connected space: Difference between revisions

Revision as of 00:01, 16 November 2015

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

The term is typically used for non-empty topological spaces.

Facts

Any topological space (not necessarily connected) can be partitioned into its connected components. The space is connected iff it has a single connected component, namely the whole space itself.

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

View other homotopy-invariant properties of topological spaces OR view all properties of topological spaces

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.
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in topology

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subspace-hereditary property of topological spaces	No	connectedness is not hereditary	It is possible to have a connected space $X$ and a subset $A$ of $X$ such that $A$ is not connected in the subspace topology.
weakly hereditary property of topological spaces	No	connectedness is not weakly hereditary	It is possible to have a connected space $X$ and a closed subset $A$ of $X$ such that $A$ is not connected in the subspace topology.
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 $\tau$ , it remains connected when we pass to a coarser topology than $\tau$ .
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.
connected union-closed property of topological spaces	Yes	connectedness is connected union-closed
closure-preserved property of topological spaces	Yes	connectedness is closure-preserved	Suppose $A$ is a subset of $X$ that is connected in the subspace topology. Then, the closure ${\overline {A}}$ is also connected in its subspace topology.

Relation with other properties

Stronger properties

Property	Meaning	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
irreducible space				\|FULL LIST, MORE INFO
ultraconnected space				\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
space with finitely many connected components				\|FULL LIST, MORE INFO
space with finitely many quasicomponents				Space with finitely many connected components\|FULL LIST, MORE INFO
space in which all connected components are open				Space with finitely many connected components\|FULL LIST, MORE INFO
space in which the connected components coincide with the quasicomponents				Space in which all connected components are open\|FULL LIST, MORE INFO

Relation with size of space

Combining connectedness with a separation axiom usually yields a lower bound on the cardinality of the space as long as it has at least two points. Below are some examples of such facts:

Other property	What its combination with being connected gives us if it has at least two points	Proof
T1 space	infinite space. In fact, any finite T1 space must be discrete	connected and T1 with at least two points implies infinite
regular space	uncountable space	connected and regular with at least two points implies uncountable
Urysohn space	uncountable space, cardinality at least that of the continuum	connected and Urysohn with at least two points implies cardinality at least that of the continuum
normal space	uncountable space, cardinality at least that of the continuum	connected and normal with at least two points implies cardinality at least that of the continuum

Opposite properties

Totally disconnected space

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 20: / Line 20: @@
 {{basicdef}}
+==Metaproperties==
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[dissatisfies metaproperty::subspace-hereditary property of topological spaces]] || No || [[connectedness is not hereditary]] || It is possible to have a connected space <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is not connected in the subspace topology.
+|-
+| [[dissatisfies metaproperty::weakly hereditary property of topological spaces]] || No || [[connectedness is not weakly hereditary]] || It is possible to have a connected space <math>X</math> and a closed subset <math>A</math> of <math>X</math> such that <math>A</math> is not connected in the subspace topology.
+|-
+| [[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]].
+|-
+| [[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]].
+|-
+| [[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>.
+|-
+| [[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.
+|-
+| [[satisfies metaproperty::connected union-closed property of topological spaces]] || Yes || [[connectedness is connected union-closed]] ||
+|-
+| [[satisfies metaproperty::closure-preserved property of topological spaces]] || Yes || [[connectedness is closure-preserved]] || Suppose <math>A</math> is a subset of <math>X</math> that is connected in the subspace topology. Then, the closure <math>\overline{A}</math> is also connected in its subspace topology.
+|}
 ==Relation with other properties==
@@ Line 72: / Line 95: @@
 * [[Totally disconnected space]]
-==Metaproperties==
-{| class="sortable" border="1"
-! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
-|-
-| [[dissatisfies metaproperty::subspace-hereditary property of topological spaces]] || No || [[connectedness is not hereditary]] || It is possible to have a connected space <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is not connected in the subspace topology.
-|-
-| [[dissatisfies metaproperty::weakly hereditary property of topological spaces]] || No || [[connectedness is not weakly hereditary]] || It is possible to have a connected space <math>X</math> and a closed subset <math>A</math> of <math>X</math> such that <math>A</math> is not connected in the subspace topology.
-|-
-| [[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]].
-|-
-| [[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]].
-|-
-| [[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>.
-|-
-| [[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.
-|-
-| [[satisfies metaproperty::connected union-closed property of topological spaces]] || Yes || [[connectedness is connected union-closed]] ||
-|-
-| [[satisfies metaproperty::closure-preserved property of topological spaces]] || Yes || [[connectedness is closure-preserved]] || Suppose <math>A</math> is a subset of <math>X</math> that is connected in the subspace topology. Then, the closure <math>\overline{A}</math> is also connected in its subspace topology.
-|}
 ==References==