Path-connected space

A topological space in which any two points can be joined by a continuous image of a simple arc; that is, a space $ X $ for any two points $ x _ {0} $ and $ x _ {1} $ of which there is a continuous mapping $ f : I \rightarrow X $ of the unit interval $ I = [ 0 , 1 ] $ such that $ f ( 0) = x _ {0} $ and $ f ( 1) = x _ {1} $. A path-connected Hausdorff space is a Hausdorff space in which any two points can be joined by a simple arc, or (what amounts to the same thing) a Hausdorff space into which any mapping of a zero-dimensional sphere is homotopic to a constant mapping. Every path-connected space is connected (cf. Connected space). A continuous image of a path-connected space is path-connected.

Path-connected spaces play an important role in homotopic topology. If a space $ X $ is path-connected and $ x _ {0} , x _ {1} \in X $, then the homotopy groups $ \pi _ {n} ( X , x _ {0} ) $ and $ \pi _ {n} ( X , x _ {1} ) $ are isomorphic, and this isomorphism is uniquely determined up to the action of the group $ \pi _ {1} ( X , x _ {0} ) $. If $ p : E \rightarrow B $ is a fibration with path-connected base $ B $, then any two fibres have the same homotopy type. If $ p : E \rightarrow B $ is a weak fibration (a Serre fibration) over a path-connected base $ B $, then any two fibres have the same weak homotopy type.

The multi-dimensional generalization of path connectedness is $ k $-connectedness (connectedness in dimension $ k $). A space $ X $ is said to be connected in dimension $ k $ if any mapping of an $ r $-dimensional sphere $ S ^ {r} $ into $ X $, where $ r \leq k $, is homotopic to a constant mapping.

References

[1]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)

Comments

A connected space is not necessarily path-connected. It is not true that in an arbitrary path-connected space any two points can be joined by a simple arc: consider the two-point Sierpinski space $ \{ 0, 1 \} $ in which $ \{ 0 \} $ is open and $ \{ 1 \} $ is not. The mapping $ f: I \rightarrow \{ 0, 1 \} $ defined by

$$ f ( x) = \left \{ \begin{array}{ll} 0 & \textrm{ if } x < 1/2 , \\ 1 & \textrm{ if } x \geq 1/2 , \\ \end{array} \right .$$

is continuous and connects 0 and 1. A space in which any two points can be joined by a simple arc is called arcwise connected. Thus, path-connected Hausdorff spaces are arcwise connected.

References

[a1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[a2]	B. Gray, "Homotopy theory. An introduction to algebraic topology" , Acad. Press (1975) pp. 15ff, 130