Inspired  by the concept in the following post
https://mathoverflow.net/questions/130797/what-are-these-compact-sets-called
We introduce the following concept:


Let $A$ be  a unital $C^{*}$ algebra. We consider the unitary equivalent relation on the set of all projections in $A$. We denote by $n(A)$, the number of equivalents classes of projections in $A$. The number of connected components of $A$ is denoted by $ncc(A)$ and is defined by $ncc(A)=log_{2} n(A)$. For  a  non unital $C^{*}$ algebra $A$, we define $ncc(A)=ncc(M(A))$ where $M(A)$ is  the multiplier algebra of $A$. This  is  a  natural definition because for  a locally compact Hausdorff   space $X$ we have $ncc(C_{0}(X))=$ The number of  connected components of $X$.



The concept in the above post is  a motivation to ask: 

**Question 1:**  Assume that We have  an extension $0\to A\to B\to C\to 0$   of $C^{*}$ algebras with $ncc(A)< \infty$  and $ncc(C)< \infty$. Does this  implies that $ncc(B)$ is  finite, too? How can $ncc(B)$ be  controlled  by $ncc(A)$  and  $ncc(C)$?


**Question 2:** How can $ncc(A\otimes_{min} B)$ be  controlled in term of $ncc(A)$  and  $ncc(B)$?  When $A$ is  commutative, is  it true  to  say that $ncc(A\otimes B) \leq ncc(A).ncc(B)$?


**Note:**  The  commutative  case gives us enough  motivation  for  such noncommutative   questions.