TITLE: I want to know why $\omega \neq \omega+1$.
QUESTION [13 upvotes]: In Kunen's book, Set Theory,chapter I.7, he said: $1+\omega=\omega \neq \omega+1$.
I want to know why $\omega \neq \omega+1$.

REPLY [4 votes]: $\omega + 1$ has a limit point (i.e. $\omega$ — using the von Neumann definition $\omega + 1 = \omega \cup \lbrace\omega\rbrace$) in the order topology while $\omega$ is discrete in the order topology.