**Theorem:**

If , , , , Hausdorff and locally compact, then there is a natural equivalence defined by , where if is a map then is given by .

We need the following two propositions in order to prove the theorem.

**Proposition**

\label{prop13}

The exponential function induces a continuous function which is a homeomorphism if and are Hausdorff and is locally compact\footnote{every point of has a compact neighborhood}.

**Proposition**

\label{prop8}

If is an equivalence relation on a topological space and is a homotopy such that each stage factors through , i.e.\ , then induces a homotopy such that .

**Proof of Theorem**

i) is surjective: Let . From Proposition \ref{prop13} we have that is a homeomorphism. Hence the function defined by is continuous since and thus . By the universal property of the quotient, defines a map such that . Thus , so that .

ii) is injective: Suppose are two maps such that , i.e.\ . Let be the homotopy rel . By Proposition \ref{prop13} the function defined by is continuous. This is because so that , thus where is a homeomorphism. For each we have . This is because if , then or . If , then . If , as is the homotopy rel . Then by Proposition \ref{prop8} there is a homotopy rel such that . Thus and similarly . Thus via the homotopy .

**Loop space**

If , we define the loop space of to be the function space with the constant loop ( for all ) as base point.

**Suspension**

If , we define the suspension of to be the smash product of with the 1-sphere.

**Corollary (Natural Equivalence relating and )**

If , and is Hausdorff, then there is a natural equivalence