Thread: Union of open sets proof

Problem: Prove that the union of any collection of open sets is open.

Is the following proof acceptable?

Proof: Let U' denote the collection of open sets and let U' = $\displaystyle \bigcup U_n$ , where n = 1,...,k. Since U' is a collection of open sets, there exists an interior point $\displaystyle s_0$ that belongs to U'. Since $\displaystyle s_0$ belongs to U', it is implied that $\displaystyle s_0$ belongs to at least one $\displaystyle U_n$. So for $\displaystyle s_0$ that belongs to $\displaystyle U_1 \in U' $, there exists a neighborhood in $\displaystyle U_1$; For $\displaystyle s_0$ that belongs to $\displaystyle U_2 \in U' $, there exists a neighborhood in $\displaystyle U_2$,...If we take the union of all $\displaystyle U_n$ , there exists a neighborhood in U' for all $\displaystyle s_0$. Therefore, U' is open.

Originally Posted by MissMousey

Problem: Prove that the union of any collection of open sets is open. Is the following proof acceptable?
Proof: Let U' denote the collection of open sets and let U' = $\displaystyle \bigcup U_n$ , where n = 1,...,k. Since U' is a collection of open sets, there exists an interior point $\displaystyle s_0$ that belongs to U'. Since $\displaystyle s_0$ belongs to U', it is implied that $\displaystyle s_0$ belongs to at least one $\displaystyle U_n$. So for $\displaystyle s_0$ that belongs to $\displaystyle U_1 \in U' $, there exists a neighborhood in $\displaystyle U_1$; For $\displaystyle s_0$ that belongs to $\displaystyle U_2 \in U' $, there exists a neighborhood in $\displaystyle U_2$,...If we take the union of all $\displaystyle U_n$ , there exists a neighborhood in U' for all $\displaystyle s_0$. Therefore, U' is open.

How do we know what is acceptable or not?
I am puzzled by this problem. Topologies are collections of open sets that are closed under arbitrary unions. Therefore, your notes must start at some other point. A neighborhood-space may be?
So unless you list definitions and axioms there is no way to comment.

my guess is that the thread-starter's definition of open is: a set U is open, if for every point x in U, there is a neighborhood of x that is a subset of U; or, equivalently, that every point of U is an interior point.

Originally Posted by Deveno

the thread-starter's definition of open is: a set U is open, if for every point x in U, there is a neighborhood of x that is a subset of U; or, equivalently, that every point of U is an interior point.

I understand that. The difficulty with that neighborhood-spaces are widely varied from R L Moore's developable regions to Helen Cullen's neighborhood relation.
In other words, what is a neighborhood?