# Path-connected space

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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 $\gamma(1) = b$.

## 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. MunkresMore info, Page 155 (formal definition)
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. ThorpeMore info, Page 52 (formal definition): introduced under name arcwise connected space
• An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. RotmanMore info, Page 25 (formal definition)