Path-connected space: Difference between revisions

Latest revision as of 19:05, 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 is a variation of connectedness. View other variations of connectedness

This article is about a basic definition in topology.
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Definition

Symbol-free definition

A topological space is said to be path-connected or arc-wise connected if given any two points on the topological space, there is a path (or an arc) starting at one point and ending at the other.

Definition with symbols

A topological space $X$ is said to be path-connected or arc-wise connected if for any two points $a,b\in X$ there is a continuous map $\gamma :[0,1]\to X$ such that $\gamma (0)=a$ and $\gamma (1)=b$ .

Relation with other properties

Stronger properties

Weaker properties

Connected space: For full proof, refer: Path-connected implies connected

Metaproperties

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

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 155 (formal definition)
Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 52 (formal definition): introduced under name arcwise connected space
An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. Rotman^{More info}, Page 25 (formal definition)

@@ Line 1: / Line 1: @@
+{{homotopy-invariant topospace property}}
 {{variationof|connectedness}}
+{{basicdef}}
 ==Definition==
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 ===Symbol-free definition===
-A [[topological space]] is said to be '''path-connected''' or ''arc-wise connected''' if given any two points on the topological space, there is a [[path]] (or an '''arc''') starting at one point and ending at the other.
+A [[topological space]] is said to be '''path-connected''' or '''arc-wise connected''' if given any two points on the topological space, there is a [[path]] (or an '''arc''') starting at one point and ending at the other.
 ===Definition with symbols===
-A [[topological space]] <math>X</math> is said to be '''path-connected''' if for any two points <math>a,b \in X</math> there is a continuous map <math>\gamma:[0,1] \to X</math> such that <math>\gamma(0) = a</math> and <math>\gamma(1) = b</math>.
+A [[topological space]] <math>X</math> is said to be '''path-connected''' or '''arc-wise connected''' if for any two points <math>a,b \in X</math> there is a continuous map <math>\gamma:[0,1] \to X</math> such that <math>\gamma(0) = a</math> and <math>\gamma(1) = b</math>.
 ==Relation with other properties==
+===Stronger properties===
+* [[Simply connected space]]
+* [[Contractible space]]
 ===Weaker properties===
@@ Line 19: / Line 27: @@
 ==Metaproperties==
-{{DP-closed}}
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
-A direct product of path-connected spaces is path-connected. This is true both for finite and infinite direct products (using the product topology for infinite direct products).
+|-
+| [[dissatisfies metaproperty::subspace-hereditary property of topological spaces]] || No || [[path-connectedness is not hereditary]] || It is possible to have a path-connected space <math>X</math> and a subset <math>A</math> of <math>X</math> such that <math>A</math> is not path-connected in the subspace topology.
-{{coarsening-preserved}}
+|-
+| [[dissatisfies metaproperty::weakly hereditary property of topological spaces]] || No || [[path-connectedness is not weakly hereditary]] || It is possible to have a path-connected space <math>X</math> and a closed subset <math>A</math> of <math>X</math> such that <math>A</math> is not path-connected in the subspace topology.
-Shifting to a [[coarser topology]] preserves the property of being path-connected. This is because a path in a finer topology continues to remain a path in a coarser topology -- we simply compose with the identity map from the finer to the coarser topology (which, by definition, must be continuous).
+|-
+| [[satisfies metaproperty::product-closed property of topological spaces]]|| Yes || [[path-connectedness is product-closed]] || Suppose <math>X_i, i \in I</math>, are all path-connected spaces. Then, the Cartesian product <math>\prod_{i \in I} X_i</math> is also a path-connected space with the [[product topology]].
-{{connected union-closed}}
+|-
+| [[dissatisfies metaproperty::box product-closed property of topological spaces]] || No || [[path-connectedness is not box product-closed]] || It is possible to have <math>X_i, i \in I</math> all path-connected spaces such that the Cartesian product <math>\prod_{i \in I} X_i</math> is ''not'' path-connected in the [[box topology]].
+|-
+| [[satisfies metaproperty::coarsening-preserved property of topological spaces]] || Yes || [[path-connectedness is coarsening-preserved]] || If <math>X</math> is path-connected under a topology <math>\tau</math>, it remains path-connected when we pass to a [[coarser topology]] than <math>\tau</math>.
+|-
+| [[satisfies metaproperty::continuous image-closed property of topological spaces]] || Yes || [[path-connectedness is continuous image-closed]] || If <math>X</math> is a path-connected space and <math>Y</math> is the image of <math>X</math> under a continuous map, then <math>Y</math> is also path-connected.
+|-
+| [[satisfies metaproperty::connected union-closed property of topological spaces]] || Yes || [[path-connectedness is connected union-closed]] ||
+|-
+| [[dissatisfies metaproperty::closure-preserved property of topological spaces]] || No || [[path-connectedness is not closure-preserved]] || It is possible to have a <math>A</math> a subset of <math>X</math> that is path-connected in the subspace topology but such that the closure <math>\overline{A}</math> is not path-connected in its subspace topology.
+|}
-A union of a family of path-connected subsets having nonempty intersection, is path-connected.
+==References==
+===Textbook references===
+* {{booklink-defined|Munkres}}, Page 155 (formal definition)
+* {{booklink-defined|SingerThorpe}}, Page 52 (formal definition): introduced under name '''arcwise connected space'''
+* {{booklink-defined|Rotman}}, Page 25 (formal definition)