Category Theory by K. H. Kamps

By K. H. Kamps

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12) Theorem Everv covering map is a fibration. For suppose that p: X ---+ X is a covering map and let IxY---X G be a homotopy lifting problem. By the general theory of covering spaces, for each Y E Y there is a unique path uy : 1---+ X such that uy(O) = f(y) and pUy(t) = G(t, y). What has to be shown is that the map (t, y) ---+ uy(t) is a continuous map F: 1 x Y ---+ X. Let y E Y. For each tEl, there exist neighborhoods U = U(t, y) of t, V = V(t, y) of y such that G(U x V) is contained in some open set in X which is evenly covered by p.

And the weak topology with respect to K is one of the standard topologies on X. If X has the weak topology with respect to a coherent family {Aa}, then X need not be a Hausdorff space even if each Aa is. Therefore X may fail to be compactly generated, even if each Aa is, just for this reason. 1) If {A a} is a coherent family of compactly generated spaces on X, and if X is a Hausdorff space (in the weak topology), then X is compactly generated. F or suppose that C is a subset of X such that C II K is closed for every compact set K.

Tn = 1 be a partition with ti - t i - I < '1 for i = 1, ... , n. Then, for each i, [ti-I, tJ c U(t;, y) for some t; and therefore [t i - I , tJ x V(t;, y) c U(t;, y) x V(t;, y) so that G([t i - I, tJ x V(t;, y)) is contained in an open set Jt; evenly covered by p. Let V(y)=n7=1 V(t;,y). 33 7 Fibrations We next define a continuous function Fy:I x V(y)~X such that (1) Fy(O, z) = J(z) and (2) pFy(t, z) = G(t, z) for z E V(y), tEl. Firstly, define Fy on {to} x V(y) by (1). Suppose that Fy has been defined on [0, tJ x V(y).

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