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## Homework Statement

Are the following regions in the plane (1) open (2) connected and (3) domains?

a. the real numbers;

b. the ﬁrst quadrant including its boundary;

c. the ﬁrst quadrant excluding its boundary;

d. the complement of the unit circle;

f. C \ Z = {z ∈ C : z [itex]\notin[/itex]Z}.

## The Attempt at a Solution

a) Open, connected, domain - because there is no boundary, all points on the plane will be interior points of the set of real numbers, and it will clearly be a connected set. Hence a domain.

b) It would be a closed set because if you picked the boundary point, an ε-neighbourhood would have a region outside this boundary, and hence a point not in the set. However, the set would be connected, can not be a domain.

c) A closed set, and connected because you should be able to get from one point to another point in the quadrant as long as it's within the boundary. Not a domain

d) Another closed set, and it is connected as long as it's within the boundary again. Not a domain again because it's not open and connected.

f) I have no idea what (f) is referring to

Are these correct? I'm not entirely sure of these answers or their reasonings. Also, I don't understand part f.