Path-connected space

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)