Define arcwise(=path) connectedness of a set in a metric space. state a relation between arcwise connectedness and connectedness of a set.

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.

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