By J. P. May

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**Example text**

The set of all such U [f ] is a basis for a topology on E since if U [f ] and U ′ [f ′ ] are two such sets and [g] is in their intersection, then W [g] ⊂ U [f ] ∩ U ′ [f ′ ] for any open set W of B such that p[g] ∈ W ⊂ U ∩ U ′ . For u ∈ U , there is a unique [g] in each U [f ] such that p[g] = u. Thus p maps U [f ] homeomorphically onto U and, if we choose a basepoint u in U , then p−1 (U ) is the disjoint union of those U [f ] such that f ends at u. It only remains to show that E is connected, locally path connected, and simply connected, and the second of these is clear.

We have the functor k : wU −→ U , and we have the forgetful functor j : U −→ wU , which embeds U as a full subcategory of wU . Clearly U (X, kY ) ∼ = wU (jX, Y ) for X ∈ U and Y ∈ wU since the identity map kY −→ Y is continuous and continuity of maps defined on compactly generated spaces is compactly determined. Thus k is right adjoint to j. We can construct colimits and limits of spaces by performing these constructions on sets: they inherit topologies that give them the universal properties of colimits and limits in the classical category of spaces.

X i B 0 ¯ : X ×I −→ Y and We first use that A −→ X is a cofibration to obtain a homotopy h ¯ then use the right-hand pushout to see that h and h induce the required homotopy ˜ h. 2. Mapping cylinders and cofibrations Although the HEP is expressed in terms of general test diagrams, there is a certain universal test diagram. Namely, we can let Y in our original test diagram be the “mapping cylinder” M i ≡ X ∪i (A × I), which is the pushout of i and i0 . Indeed, suppose that we can construct a map r that makes the following diagram commute: i0 G A×I v v vv v vv v{ v i×id i M a i r r r || | r | r || r || G X × I.