Homotopy of paths
A homotopy of paths in a space 
is a family 
, 
, such that
(i) The endpoints 
and 
are independent of 
.
(ii) The associated map 
defined by 
is continuous.
When two paths 
and 
are connected in this way by a homotopy 
, they are said to be homotopic. The notation for this is 
.
Example: Linear Homotopies
Any two paths and in 
having the same endpoints 
and 
are homotopic via the homotopy 
Simply-connected
A space is called simply-connected if it is path-connected and has trivial fundamental group.
A space is simply-connected iff there is a unique homotopy class of paths connecting any two parts in .
Path-connectedness is the existence of paths connecting every pair of points, so we need to be concerned only with the uniqueness of connecting paths.
(
) Suppose 
. If 
and 
are two paths from to , then 
since the loops 
and 
are each homotopic to constant loops, due to 
.
(
) Conversely, if there is only one homotopy class of paths connecting a basepoint to itself, then all loops at are homotopic to the constant loop and .
is isomorphic to 
if and 
are path-connected.
A basic property of the product topology is that a map 
is continuous iff the maps 
and 
defined by 
are both continuous.
Hence a loop in 
based at 
is equivalent to a pair of loops in and 
in based at and 
respectively.
Similarly, a homotopy of a loop in is equivalent to a pair of homotopies 
and 
of the corresponding loops in and .
Thus we obtain a bijection 
, ![[f]\mapsto([g],[h])](https://s0.wp.com/latex.php?latex=%5Bf%5D%5Cmapsto%28%5Bg%5D%2C%5Bh%5D%29&bg=ffffff&fg=333333&s=0 "[f]\mapsto([g],[h])")
. This is clearly a group homomorphism, and hence an isomorphism.
Note: The condition that and are path-connected implies that 
, 
.
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