By John Hales

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Y1, ... , Yk E {E*, EW}) the following properties arc equivalent: l. e. open. 2. There is some decidable set V C;;; Y x E* such that X = {y E Y I (3u E E*)(y, 11) E V}. 3. e. open set V C;;; Y x E* such that X = {y E Y I (311 E E*)(y, u) E V}. 2). 12. Every decidable set X C;;; Y1 X ... X Yk (Yi E {E*, EW}) is recursive. There are recursive sets which are not decidable. 9). 6. 4. l. 2 2. Prove the projection theorem. 3. e. opcn" and "decidable" are replaced by "open" and "clopen", respectively.

As a canonical base of T* we consider the set { {w} Iw E E*}. 4. As a canonical ba8e of Te we con8ider the set {wEW Iw E E*}. 5. On Y := Y1 X ... X Y k (Yl, ... , Y k E {E*, EW}) we consider the product topology with the canonical base /3 y := {y 0 Y lyE (E*)k}, where (Yl, ... ,Yk) 0 Y:= U1 x ... 7* and {{ w} I w E E*} is a base of T*. 7'" I w r;;; p} E Te. 7* we have vEw n wEw = (wE'" if v r;;; w, vEw if 11J r;;; v, and otherwise). Therefore, the set f3 := {wE'" Iw E E*} is closed under finite intersection, and so it is a base of a topology.

Proof: l. Let AI be a Type-2 machine computing f :C;;; EW -t E*. vI halts after exactly otherwise. div Iwl steps Then h is computable, has prefix-free domain and satisfies f = h*. Notice that dom(h) is even recursive. On the other hand, let AI be a Turing machine computing h :C;;; E* -t E* with prefix-free domain. ;) and halts. Since dom(h) is prefix-free, there is at most one such number k. Obviously, N computes the function h*. 2. 'vI be a Type-2 machine computing f :C;;; E W -t EW. 'vI on input wOw produces in Iwl steps.