Question

this question is about complex variables

Exercise 2 (i If is open in C and Ac C any subset, then ON A is open in A (ii) If A C C is any subset and U is open in A, then there exists 2 open in C with 2NA=U. Hint. Use the definition and construct O as union of disks.

Answer 1

(i). Given $\Omega$ is a open set in $\mathbb{C}$ and be any set .

AC

Let be an arbitrary element ,

E A

E2, E A

As
ES
and $\Omega$ is open so there exist a neighbourhood of x say N_x such that $x \in N_{x} \subset \Omega$

NnAC A
is an neighbourhood of x containted in $\Omega \cap A$ .

$\Rightarrow$ x is an interior point of $\Omega \cap A$ , sincce x is arbitrary so every point of $\Omega \cap A$ is an interior point .

Hence $\Omega \cap A$ is open in A .

(ii). $A \subset \mathbb{C}$ be any set and is open in .

Let , $\Omega =$ int (A) $\cap U$ i.e., $\Omega$ is the collection of all interior point of A which is an U so $\Omega$ is an open set as it is intersection of two open set .

As U a open subset of A so each point of U is an interior point of A

$\Rightarrow$$U \subset int(A)$

Hence ,

$\Omega \cap A$

= int (A) $\cap U$$\cap A$

= U , since $U \subset A , U\subset int(A)$ .