Question

Determine whether or not the given set is (a) open, (b) connected, and (c) simply-connected.

29.

Short answer

a) The set\(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\)is not open.

b) The set\(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\)is connected.

c) The set \(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\) is simply-connected.

Step-by-step solution

Given data 

The given set is \(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\).

Condition for set to be open, connected, or simply connected

A region is open if it does not contain any of its boundary points. A region is connected if one can connect any two points in that region with a path that lies completely in that particular region. A region is simply-connected if it is connected and it contains no holes

Check whether the given set is open or not

a)

The set\(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\)possesses a semi-annual region located in upper half-plane between the circles with origin as a center and of radii 1 and 2 respectively.

Draw the set \(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\) as shown in Figure 1.

It is observed that set\(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\)is including any boundaries as shown above. Consider the boundary point\((1,0)\)the disk cannot entirely lie in the set. Hence set is not open.

Thus, the set\(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\)is not open.

Check whether the given set is connected or not

b)

Set is\(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\).

In Figure1, it is observed that set\(\{ (x,y)|1 < |x\mid < 2\} \)consists one separate part alone. Hence set is connected.

Thus, the set \(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\) is connected.

Check whether the given set is simply-connected or not

c)

Set is\(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\).

In the graph, it is observed that set\(\{ (x,y)|1 < |x\mid < 2\} \)consists one separate part alone. Hence set is connected.

Thus, the set \(\left\{ {(x,y)\mid 1 < {x^2} + {y^2} < 4,y \ge 0} \right\}\) is connected.