set & domain

[logistic_guy]

Sketch.

$\displaystyle |z - 2 + i| \leq 1$

Is this set a domain?

 

[pka]

Sketch.

$\displaystyle |z - 2 + i| \leq 1$

Is this set a domain?

The answer to this question strictly depends on which course it comes from, more especially the textbook is used.
It appears that this is a question about complex variables.
If you have access to a reasonably good library, look for Introduction To Complex Variables by F, P. Greenleaf.
That textbook defines a domain as a connected open set in the plane.
The set $\displaystyle \{z\in\mathbb{C}:|z-2+i|\le 1\}$ being the set of points( a disk) centered at $\displaystyle 2-i$ with radius $\displaystyle \le 1$ is clearly connected but also it is closed.

 

[logistic_guy]

The answer to this question strictly depends on which course it comes from, more especially the textbook is used.
It appears that this is a question about complex variables.
If you have access to a reasonably good library, look for Introduction To Complex Variables by F, P. Greenleaf.
That textbook defines a domain as a connected open set in the plane.
The set $\displaystyle \{z\in\mathbb{C}:|z-2+i|\le 1\}$ being the set of points( a disk) centered at $\displaystyle 2-i$ with radius $\displaystyle \le 1$ is clearly connected but also it is closed.

Does this mean that a set that contains all of its accumulation points is never a domain? Since it is closed!