Answer:
See definitions and relation below
Step-by-step explanation:
Given points x and y of a certain set S in a metric space, a path from x to y is a continuous map f:[a,b]-->S of some closed interval [a,b] in the real line into S, such that
f(a)=x and f(b)=y
In this case, we can also say that the points x and y are joined by a path or arc.
A set S in metric space is said to be path connected or arcwise connected if every pair of points x, y of S can be joined by a path.
The relation between arcwise connectedness and connectedness of a set is that every arcwise connected set is also connected, but the converse does not hold; not every connected space is also path connected.
As an example, consider the unit square [0,1]X[0,1] with the dictionary order topology.
It can be proved that this space is connected but not path connected.
