(a)
Whether the given set is open or not
Solution:
Open
Explanation:
1) Concept:
2) Given:
3) Calculations:
Consider the given set,
Sketch of the region
The set
That is
For points
The set
By using the concept, the set
Conclusion:
The set
(b)
Whether the given set is connected or not
Solution:
Connected
Explanation:
1) Concept:
2) Given:
3) Calculations:
Consider the given set,
Sketch of the region
The set
That is
The region
Also, any two points chosen in
By using the concept, the set
Conclusion:
The set
(c)
Whether the given set is simply connected or not.
Solution:
Not simply connected
Explanation:
1) Concept:
A simply connected region in the plane is a connected region
2) Given:
3) Calculations:
Consider the given set,
Sketch of the region
The set
That is
The region
Consider two points
See that the path joining the two points
Since it passes through the point
Conclusion:
The set
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