I learned that connected components are always closed, but path-components are not necessarily closed.
I know the following fact
1.A path-component is maximum path-connected subset (including a point in the subset)
I finally got the conclusion that I've got to find an example that $B$ is path-connected but $\bar B$ is not path-connected
can you give me an example as simple as possible?
The classic example is the so-called topologist's sine curve. It is the closure in $\Bbb R^2$ of the graph of the function $f:(0,\infty)\to \Bbb R$ given by $$ f(x)=\sin(1/x) $$ This space has two path components: the graph itself, and the line segment along the $y$-axis going from $(0,-1)$ to $(0,1)$. But the space is connected, because every open neighbourhood around a point on the line segment contains points from the graph as well. Therefore, within this space, as in $\Bbb R^2$, the closure of the graph is the whole space.
But there is no continuous path going from a point on the graph to a point on the segment. Therefore, the two are distinct path components.