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## 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