Complex Analysis: Open Set Intersection Proof

[Pyroadept]

Homework Statement

An open set in the complex plane is, by definition, one which contains a disc of positive radius about each of its points. Prove that:
(a) the intersection of two open sets is an open set
(b) the union of arbitrarily many open sets is an open set

Homework Equations

If A, B are two sets, A = {a, b, c} and B = {c, d, e}, then:
AUB = {a, b, c, d, e}
A intersect B = {b}

The Attempt at a Solution

Here's what I've done:

(a) Let U, V be two non-empty open sets in the complex plane. Then, by definition, all the points of U and V have a positive radius about them.
Then, clearly, all the points in the intersection of U and V will be points that have a positive radius about them.
Thus the intersection of U and V is, by definition, an open set.

(b) Let U_1, U_2, ... , U_n be any n non-empty open sets in the complex plane. Then, by definition, all the points of U_1, U_2, ... , U_n have a positive radius about them.
Thus, clearly, all the points in the union of U_1, U_2, ... , U_n have a positive radius about them.
Thus the union of U_1, U_2, ... , U_n is, by definition, an open set.

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I know heuristic proofs don't generally cut it in analysis, but I really don't see what else there is to say on this particular question. Am I missing something fundamental in my argument?
Thanks for any help

 

[jbunniii]

I think you need to be a lot more explicit about the positive radius.

Here's how I would start (a):

Let U and V be two open sets in \mathbb{C}. If U \cap V is empty, then the claim is obviously true. So suppose that it is not empty, and choose z \in U \cap V.

Then z \in U, so there exists a radius r_u > 0 such that u \in U whenever |z - u| < r_u.

Similarly, z \in V, so there exists a radius r_v > 0 such that v \in V whenever |z - v| < r_v.

I now wish to show that U \cap V contains a disc of positive radius around the point z, which means that I must find a radius r > 0 such that x \in U \cap V whenever |z - x| < r.

Your job is now to find such an r. Hint: will one of r_u or r_v work? If so, which one? And you must prove that it works!