Topology Question

[grand pilot]

We know that x belongs to Q is boundary point (∂ Q) by using the two definitions in Figure 2, but what the hell is this latter ∂ Q = R?

[jagged socket]

For the record, the pages were uploaded in reverse order

[jagged socket]

You are showing that the boundary of Q is R. That means, by definition, every r in R is a boundary point in Q.

Choose any point r in R and choose any e>0.

We must show that open ball of radius e and centered at r will contain both a rational and an irrational number in it. Do you know why?

[grand pilot]

Probably because on the coordinate axis, the left and right sides of a rational number are irrational numbers

[jagged socket]

Unfortunately, there isn't much sense in what you're saying, at least in the view of analysis and topology.

[grand pilot]

How would you proof that

[jagged socket]

We must show that every $r\in\mathbb{R}$ is a boundary point of $\mathbb{Q}$.

[raven oxide]

bonimy

[jagged socket]

So let's choose any $r\in\mathbb{R}$ and prove that it is a boundary point.

[raven oxide]

bonimy