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Path-connected space

Revision as of 19:05, 26 January 2012 by Vipul (talk | contribs) (Metaproperties)
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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

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)