5/5 



AN INTRODUCTION TO CYBERNETICS 



5/5. Stable region. If a is a state of equilibrium, T{a) is, as we 

 saw in S.5/3, simply a. Thus the operation of T on a has generated 

 no new state. 



The same phenomenon may occur with a set of states. Thus, 

 suppose T is the (unclosed) transformation 



abcdefgh 



p g b f a a b ni 



It has no state of equilibrium; but the set composed of b and g has 

 the pecuUarity that it transforms thus 



b g 



T:\ 



T-.i, 



g b 



i.e. the operation of T on this set has generated no new state. Such a 

 set is stable with respect to T. 



Fig. 5/5/1 



This relation between a set of states and a transformation is, of 

 course, identical with that described earlier (S.2/4) as "closure". 

 (The words "stable set" could have been used from there onwards, 

 but they might have been confusing before the concept of stability 

 was made clear; and this could not be done until other matters had 

 been explained first.) 



If the transformation is continuous, the set of states may He in a 

 connected region . Thus in Fig. 5/5/ 1 , the region within the boundary 

 A is stable; but that within B is not, for there are points within the 

 region, such as P, which are taken outside the region. 



The concept of closure, of a stable set of states, is of fundamental 

 importance in our studies. Some reasons were given in S.3/2, 



76 



