Point set proof

Dick

That's exactly what I wanted to hear. Thanks!

Zondrina

Oh man thanks so much for your patience, really though. I just wanted to understand this so badly.

Also my final concern, is what jbuni said true? Am I too close? Or was my proof sufficient?

Dick

What you said convinces me you understand it. jbunnii's rephrasing is a better version for the proof.

Zondrina

Therefore since A is contained within AUB, it follows that A⁰ is contained within (AUB)⁰.

I believe that's what he meant to say. If so then I do agree it's a better way to phrase this.

Dick

Yes, if the neighborhood N of x is in A, then it's certainly in AUB.

Zondrina

Perfect, thanks again for all your help man. Though I wish my professor didn't dive right into topology as soon as the class started... ( Its only calc II lol ).

jbunniii

Just to follow up on my earlier post...

You have a neighborhood N of x such that [itex]x \subset N \subset A[/itex]. Furthermore, [itex]A \subset A \cup B[/itex], so it follows that [itex]x \subset N \subset A \cup B[/itex]. Therefore x is an interior point of [itex]A \cup B[/itex], i.e. [itex]x \in (A \cup B)^o[/itex]. Since [itex]x[/itex] was an arbitrary point of [itex]A^o[/itex], this shows that [itex]A^o \subset (A \cup B)^o[/itex].

(Then argue similarly for [itex]x \in B^o[/itex] and wrap up the proof.)

That's the level of detail I would like to see if I were grading this problem, so I could be confident that you understood why every step was true.