RECENT ADVANCES INHOMOTOPY THEORY 3 natural homeomorphism* Let (X, A) b e a n admissibl e pai r i n S Q , an d le t p : X - » A7'/ 4 b e th e identificatio n map, i: A .— A th e inclusio n map . The n i A 1: A AY I A l 7 i s a n imbedding , s o that w e ma y regar d A AY a s a subspac e o f X A Y\ Moreover , p A 1 : X AY •— AV/4 A Y send s / I A Y int o th e base-point , an d w e hav e (1.5) The map p A 1 induces a homeomorphism of X A Y/A A Y with (X/A) A Y. In dealing wit h functio n spaces , w e shal l assum e th e domain s t o b e compact . Th e reason fo r thi s i s tha t th e categor y b i s no t know n t o admi t functio n spaces , eve n though th e categor y o f compactl y generate d space s does . However , i f A 7 i s compac t and Y 6 § 0 , the n i t follow s fro m th e result s o f Steenro d [68 ] an d Milno r [55 ] tha t th e space F{X, Y) o f al l map s o f X int o Y wit h th e compact-ope n topolog y doe s belon g to s 0 . Let £ Q b e th e ful l categor y o f compac t space s i n § Q . (Not e that , i f X 6 £ the n X i s dominate d b y a finite complex I do no t kno w whethe r th e convers e i s true. ) W e then hav e (1.6) If X,Y e e , Z S , then F{X A Y, Z ) and F{X, FiY, Z) ) are naturally homeomorphic. (1-7) If X £ , y b then the evalution map e: F{X, Y) A X -— » Y is continuous. Let / b e th e fre e uni t interval , T th e uni t interva l wit h base-poin t 0 , T ~ S th e subspace [0 , l i o f T, S - S - T/T. W e the n hav e th e cone , suspension , path , an d loop functors , define d b y TX=TA X, SX = S A X, PX = F{T, X), SIX = F(S, X). Moreover (1.8) The space S A X is naturally homeomorphic with X and SX with TX/S° AX = TX/X. (1-9) The spaces SX A Y and X A SY are naturally homeomorphic. (1.10) If Xe £ 0 , the spaces F(SX, Y) , F{X, QY ) and QF(A \ Y ) are naturally homeomorphic. Let [A , Y ] b e th e set o f homotop y classe s o f map s o f X into Y . Wit h th e opera - tion o f two-side d composition , w e obtai n a functo r [ , ] fro m b x b t o th e categor y of set s wit h base-point thi s functo r i s contravarian t i n it s firs t argument , covarian t
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