View source for Path-connected space

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A [[Topological space|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|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|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|homotopy type]]. If  $  p :  E \rightarrow B $
is a weak fibration (a [[Serre fibration|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>

====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 \{

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