Give an example of a refinement in mathematical modeling with contextual explanation.

Consider the set [0, 1], and consider it's partition $A=(0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1)$ and consider a refinement of this partition, B Note that refinement of a partition A contains all points of A and possibly some other points, thus B can be considered as: $B=(0,\frac{1}{8},\frac{1}{4},\frac{1}{2},\frac{3}{4},\frac{7}{8},1)$ Here it can be seen that A is contained in B, i.e. A sub B And if we consider another refinement of A: $B_1=(0,\frac{1}{8},\frac{1}{4},\frac{3}{8},\frac{1}{2},\frac{3}{4},1),\text{ again note that }A\text{ sub }{B}_1$ i.e. $B_1$
is an refinement of $A\text{ but B}\text{⧸}\subset B1\to \text{ sub }{B}_1$ is not an refinement of B.