Question

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

30. \(\{ (x,y)\mid (x,y) \ne (2,3)\} \)

Short answer

a) The set\(\{ (x,y)\mid (x,y) \ne (2,3)\} \)is open.

b) The set\(\{ (x,y)\mid (x,y) \ne (2,3)\} \)is connected.

c) The set \(\{ (x,y)\mid (x,y) \ne (2,3)\} \) is not simply-connected

Step-by-step solution

Given data

The given set is \(\{ (x,y)\mid (x,y) \ne (2,3)\} \).

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\(\{ (x,y)\mid (x,y) \ne (2,3)\} \)comprises of all the points in x y-plane except points(2,3).

Draw the set \(\{ (x,y)\mid (x,y) \ne (2,3)\} \) as shown in Figure 1.

It is observed that set\(\{ (x,y)\mid (x,y) \ne (2,3)\} \)consists of only one boundary point\((2,3)\)and that is not included. Hence set is open.

Thus, the set \(\{ (x,y)\mid (x,y) \ne (2,3)\} \) is open.

Check whether the given set is connected or not 

b)

In the graph, it is observed that set\(\{ (x,y)\mid (x,y) \ne (2,3)\} \)consists one separate part alone. Hence set is connected.

Thus, the set \(\{ (x,y)\mid (x,y) \ne (2,3)\} \) is connected.

Check whether the given set is simply-connected or not

c)

The set\(\{ (x,y)\mid (x,y) \ne (2,3)\} \)consists a hole at\((2,3)\). therefore, consider a simple curve in a such a way that encloses\((2,3)\). The curve includes a point which in not located in set\(\{ (x,y)\mid (x,y) \ne (2,3)\} \). Hence it is not simply-connected.

Thus, the set \(\{ (x,y)\mid (x,y) \ne (2,3)\} \) is not simply-connected