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