182 [5.9]TOPOLOGY AND GROUPOIDS(i) Fig. 5.11(ii)Strangely, the general case of this problem has proved not entirely tractable within the traditional outlook of general topology, that is, using topological spaces and continuous functions. In this section we describe one method of dealing with this difulty. A space X is called a k-space if X has the al topology with respect to all maps C 鈫�X for all compact Hausdorff spaces C. In such case, a function f : X 鈫�Y is continuous if and only if ft : C 鈫�Y is continuous for all compact Hausdorff spaces C and maps t : C 鈫�X. (The use of the letter k is traditional. It was introduced because the German for compact is kompakte.) At st sight this deition seems ridiculous, since a property of a space X is described by reference to a large class of spaces. However, as should be clear from earlier chapters, this procedure is in the modern spirit and can be convenient precisely because of this global reference. The following result shows however that in order to test for a k-space we need look only at a set of test spaces C. 5.9.1 Let X be a space. Then the following are equivalent: (a) X is a k-space; (b) there is a set CX of maps t : Ct 鈫�X for compact Hausdorff spaces Ct such that a set A is closed in X if and only if t� (A) is closed in Ct for all t �CX ; (c) X is an identiation space of a space which is a sum of compact Hausdorff spaces. Proof (a) 鈬�(b) The set CX is constructed as follows. Since X is a k-space, for each non-closed subset B of X there is a compact Hausdorff space CB and map t : CB 鈫�X such that t� [B] is not closed in CB . Choose one such CB and one such t for each non-closed B, and let CX be the set of all these t. That this set has the required property is clear. (b) 鈬�(c) Suppose CX given as in (b). Let K be the sum of the spaces Ct for all t �CX , and let it : Ct 鈫�K be the inclusion. Let p : K 鈫�X be the
