Provide an example showing that the path components of a topological spaces don't need to be open nor closed sets in the space.
Since connectedness components are always both open and close sets, you need an example in which "connected" is not the same as "path connected". Let A be the line segment, in , from (-1, 0) to (0,0) inclusive (so it is a closed set). Let B be the graph of y= sin(1/x) for . Let the topological space, X, be . A and B are "path components" of X but B is neither open nor closed.