Long line is connected and compact

Question

How to prove that the long line is connected and compact. I was trying to prove connectedness using contradiction but couldn't.

Answer 1

You can actually show that the long line is path-connected, which shows that it is connected. Pick any two points $x=\langle\alpha,s\rangle$ and $y=\langle\beta,t\rangle$ on the long line, with $x<y$. If $\alpha=\beta,$ then $s<t$ and the long line interval $[x,y]_L$ is readily homeomorphic to the real interval $[s,t]_{\Bbb R}$, so $x,y$ are connected by a path. Otherwise, $[x,y]_L$ is the disjoint union of (in increasing order) $\alpha\times[s,1)$, then a well-ordered (at most) countably-infinite set of intervals of the form $\gamma\times[0,1)$ with $\alpha<\gamma<\beta,$ then $\beta\times[0,t],$ joined end to end. You should again be able to show an explicit homeomorphism with a closed real interval, so that $x,y$ are connected by a path. I leave the finite case to you.

As a hint for how to tackle the infinite case, suppose that $A$ is a countably-infinite set well-ordered by $\prec,$ with $s(a)$ the immediate successor of $a$ in $\langle A,\prec\rangle$ for any non-maximum $a,$ and that $f:À\to\Bbb Z^+$ is any bijection. A ready proof by recursion shows that for all $a\in A,$ $$g(a)=1+\sum_{b\prec a}2^{-f(b)}$$ is an element of $[1,2)_{\Bbb R},$ so that $g:A\to [1,2)_{\Bbb R}$ is a function. Moreover, $g$ is readily an increasing function, and it can be shown that for all $c<2,$ there is some $a\in A$ such that $c<g(a).$ Thus, $$[0,1)_{\Bbb R}\cup\left(\bigcup_{a\in A}\Bigl[g(a),g\bigl(s(a)\bigr)\Bigr)_{\Bbb R}\right)\cup[2,3]_{\Bbb R}=[0,3]_{\Bbb R}.$$ Using these facts, one can define an explicit bijection $[x,y]_L\to[0,3]_{\Bbb R}.$

However, the long line is not compact. For example, let $z$ be the deleted least element of the long line, and for each $\alpha<\omega_1,$ let $z_\alpha=\langle\alpha,0\rangle.$ Then the set of long line intervals $(z,z_\alpha)_L$ forms an open cover of the long line with no finite subcover (possibly not even a countable subcover, but that is not known). It is, however, locally compact, as every element is contained in a non-degenerate closed long line interval homeomorphic to a closed and bounded real interval.