Thread: Open sets Proof

A) Show that the union of two nonempty open sets is open.

I have this theorem to use: A set D is open iff it contains no point of its boundary.

Can someone show have to give a solid proof of A...

Thanks in advance for any help

Here is what I think:

Say I have open sets X and Y, then for , I can find an open ball centered at them such that the entire ball is in X and Y, respectively.

Well, then, if you pick any point in , that that point is either in X or Y, well, then, you can draw another open ball that is contained in either X or Y.

So if you use radius for the points in X and Y, you can just use radius

Hope this helps.

Originally Posted by jzellt

A) Show that the union of two nonempty open sets is open.

I have this theorem to use: A set D is open iff it contains no point of its boundary.

Can someone show have to give a solid proof of A...

Thanks in advance for any help

Is this a topological space or a metric space? If it is the former than this is the definition. Otherwise, let be an arbitrary class of open sets in a metric space . Let be arbitrary. Since for some and is open there exists a such that and we are finished.