Motivated  by  classical  Gelfand   Naimark  duality, the  correspondence  between  the category of  commutative  $C^{*}$  algebras  and the  category of  locally  compact  Hausdorff spaces,  we  define  a  "continum algebra",  as  follows:

A  "continum  algebra" is a  (separable)  unital algebra  without  non trivial idempotent. The reason  for  this  definition is  that  for  a  classical space $X$, the algebra $C(X)$ satisfies the above  conditions if  and  only  if  $X$ is  a  continum.

Motivated  [by this  post](http://mathoverflow.net/questions/259017/two-consecutive-continua), we  have  the following definition:


A pair  $(A,B)$ of continum $C^{*}$ algebras is  a "Consecutive" pair if there  is  a  surjective  morphism $\phi:A \to B$  such that for  every  continum algebra  $C$  and  surjective  morphisms  $A\to C \to  B$ we  have  either $C \simeq A, \text{or}\;\;  C \simeq B$.

A  consecutive  resolution of  $A$ to a simple  algebra $A_{0}$ is  a  finite  sires  of continum  algebras $A=A_{n}, A_{n-1}, A_{n-2},\ldots,A_{0}$  such that  every  pair $(A_{i}, A_{i-1})$  is  a  consecutive  pair.

>Question: To  what  extent,  all  continum algebras which  possess such  resolutions, are  classified? Is there  an already  knowm name  for  such  algebras?  Does  every  commutative  algebra $A=C(M)$  admit such  resolution, where $M$  is  a  compact  manifold?

 The  later can be  restated as  follows:

>Assume that $M$ is  a  compact  manifold: Are there compact  connected  subsets $M=X_{n}\supset  X_{n-1}\supset  \ldots\supset X_{0}=\{pt\} $  such that each  pair $(X_{i},X_{i-1})$  is  a  [consecutive  pair](http://mathoverflow.net/questions/259017/two-consecutive-continua)?