Question

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

Answer 1

Arcwise connectedness is defined as the presence of a continuous path between any two points in a set in a metric space. There is a relation between arcwise connectedness and connectedness, where any path connected set is also connected, but the converse is not necessarily true.

Step-by-step explanation:

A set in a metric space is said to be arcwise connected or path connected if there exists a continuous curve or path that connects any two points in the set.

The relation between arcwise connectedness and connectedness of a set is that any arcwise connected set is also connected, but the converse is not necessarily true. In other words, every path connected set is connected, but not every connected set is path connected.

For example, consider a set consisting of two separate points in a metric space. This set is connected because we cannot find two disjoint open sets that cover the set, but it is not arcwise connected because there is no continuous path connecting the two points.

Answer 2

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.