**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 , . This is clearly a group homomorphism, and hence an isomorphism.

Note: The condition that and are path-connected implies that , .

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