The Katětov Space

Question

The Katětov Space defined as $\Lambda=\beta \mathbb{R}-(\beta \mathbb{N}-\mathbb{N})$.

I don't know why $\mathbb{R}\subset \Lambda$ and hence $\beta \Lambda=\beta \mathbb{R}$.

Someone can help me?

Answer 1

I recall the following basic facts about the Čech-Stone compactification from Ryszard Engelking’s “General Topology” (2nd ed., Heldermann, Berlin, 1989).

As I understood, $\beta\Bbb N\subset\beta\Bbb R$ is the closure $\overline{\Bbb N}$ of $\Bbb N\subset\Bbb R $ in $\beta\Bbb R$. A set $\overline{N}\cap\Bbb R$ is the closure in the space $\Bbb R$ of its subspace $\Bbb N$. Since $\Bbb N$ is closed in $\Bbb R$, $\overline{\Bbb N}\cap\Bbb R=\Bbb N$. Then $$\Bbb R\subset(\beta\Bbb R\setminus\overline{\Bbb N})\cup (\overline{\Bbb N}\cap \Bbb R)= (\beta\Bbb R\setminus\overline{\Bbb N})\cup {\Bbb N}=\Lambda.$$

Therefore $\beta \Lambda=\beta \mathbb{R}$ by Corollary 3.6.9.