Re: Simple Topology Proof

Re: Simple Topology Proof

Hmmm... So... Just do a similar construction?

Since belong to , then either or , which guarantees that there is a leftmost point in .

Re: Simple Topology Proof

Quote:

Originally Posted by

**Aryth** Hmmm... So... Just do a similar construction?

Since

belong to

, then

either

or

, which guarantees that there is a leftmost point in

.

**No you are not done.**

There is only __one__ *left most* point of

Now it is one of Now we know that .

Finish it!

Re: Simple Topology Proof

I'm slightly confused... I know what you're saying, but I can't see the rest of the proof... I mean, I could suppose that and that would make the left most point, or I could say the opposite, and would be the left most point. I feel like I'm missing something.

Re: Simple Topology Proof

Quote:

Originally Posted by

**Aryth** I'm slightly confused... I know what you're saying, but I can't see the rest of the proof... I mean, I could suppose that

and that would make

the left most point, or I could say the opposite, and

would be the left most point. I feel like I'm missing something.

You know .

Take each case. Prove that is the *left most point.*.

Re: Simple Topology Proof

Ok...

Note: If , then . In any of these cases, either or is the left most point.

We know:

Let , then they are the same point, and that point is the left most point, since it is also the left most point of both and .

Let , then and , . In other words, it means that is to the left of all points in , and is therefore the left most point of the union.

Let , then and , . In other words, it means is to the left of all points in , and is therefore the left most point of the union.

Re: Simple Topology Proof

Well done. It is now complete.

I think it is generally true that topologist are sticklers for completeness in proofs .

Re: Simple Topology Proof

I really appreciate the help. And I'll keep that in mind, I'm getting that feeling from my professor.