LongLine 2003-02-27 04:39:53 2007-07-18 21:29:29 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} %--------------------- Greek letters, etc ------------------------- \newcommand{\CA}{\mathcal{A}} \newcommand{\CC}{\mathcal{C}} \newcommand{\CM}{\mathcal{M}} \newcommand{\CP}{\mathcal{P}} \newcommand{\CS}{\mathcal{S}} \newcommand{\BC}{\mathbb{C}} \newcommand{\BN}{\mathbb{N}} \newcommand{\BR}{\mathbb{R}} \newcommand{\BZ}{\mathbb{Z}} \newcommand{\FF}{\mathfrak{F}} \newcommand{\FL}{\mathfrak{L}} \newcommand{\FM}{\mathfrak{M}} \newcommand{\Ga}{\alpha} \newcommand{\Gb}{\beta} \newcommand{\Gg}{\gamma} \newcommand{\GG}{\Gamma} \newcommand{\Gd}{\delta} \newcommand{\GD}{\Delta} \newcommand{\Ge}{\varepsilon} \newcommand{\Gz}{\zeta} \newcommand{\Gh}{\eta} \newcommand{\Gq}{\theta} \newcommand{\GQ}{\Theta} \newcommand{\Gi}{\iota} \newcommand{\Gk}{\kappa} \newcommand{\Gl}{\lambda} \newcommand{\GL}{\Lamda} \newcommand{\Gm}{\mu} \newcommand{\Gn}{\nu} \newcommand{\Gx}{\xi} \newcommand{\GX}{\Xi} \newcommand{\Gp}{\pi} \newcommand{\GP}{\Pi} \newcommand{\Gr}{\rho} \newcommand{\Gs}{\sigma} \newcommand{\GS}{\Sigma} \newcommand{\Gt}{\tau} \newcommand{\Gu}{\upsilon} \newcommand{\GU}{\Upsilon} \newcommand{\Gf}{\varphi} \newcommand{\GF}{\Phi} \newcommand{\Gc}{\chi} \newcommand{\Gy}{\psi} \newcommand{\GY}{\Psi} \newcommand{\Gw}{\omega} \newcommand{\GW}{\Omega} \newcommand{\Gee}{\epsilon} \newcommand{\Gpp}{\varpi} \newcommand{\Grr}{\varrho} \newcommand{\Gff}{\phi} \newcommand{\Gss}{\varsigma} \def\co{\colon\thinspace} \PMlinkescapeword{induced} The \emph{long line} is a non-paracompact Hausdorff $1$-dimensional manifold constructed as follows. Let $\GW$ be the first uncountable ordinal (viewed as an ordinal space) and consider the set $$L:=\GW\times [0,1)$$ endowed with the order topology induced by the lexicographical order, that is the order defined by $$(\Ga_1,t_1) < (\Ga_2,t_2) \iff \Ga_1<\Ga_2 \quad\text{or}\quad (\Ga_1=\Ga_2 \quad\text{and}\quad t_1<t_2)\,.$$ Intuitively $L$ is obtained by ``filling the gaps'' between consecutive ordinals in $\GW$ with intervals, much the same way that nonnegative reals are obtained by filling the gaps between consecutive natural numbers with intervals. Some of the properties of the long line: \begin{itemize} \item $L$ is a chain. \item $L$ is not compact; in fact $L$ is not Lindel\"of. Indeed $\left\{\,[\,0,\Ga):\Ga<\GW\right\}$ is an open cover of $L$ that has no countable subcovering. To see this notice that $$\bigcup\left\{\, [\,0,\Ga_x):x\in X\right\}=\left[\,0,\sup\{\Ga_x:x\in X\}\right)\,$$ and since the supremum of a countable collection of countable ordinals is a countable ordinal such a union can never be $[\,0,\GW)$. \item However, $L$ is sequentially compact. Indeed every sequence has a convergent subsequence. To see this notice that given a sequence $a:=(a_n)$ of elements of $L$ there is an ordinal $\Ga$ such that all the terms of $a$ are in the subset $[\,0,\Ga\,]$. Such a subset is compact since it is homeomorphic to $[\,0,1\,]$. \item $L$ therefore is not metrizable. \item $L$ is a $1$--dimensional locally Euclidean \item $L$ therefore is not paracompact. \item $L$ is first countable. \item $L$ is not separable. \item All homotopy groups of $L$ are trivial. \item However, $L$ is not contractible. \end{itemize} \section*{Variants} There are several variations of the above construction. \begin{itemize} \item Instead of $[\,0,\GW)$ one can use $(0,\GW)$ or $[\,0,\GW\,]$. The latter (obtained by adding a single point to $L$) is compact. \item One can consider the ``double'' of the above construction. That is the space obtained by gluing two copies of $L$ along $0$. The resulting open manifold is not homeomorphic to $L\setminus \{0\}$. \end{itemize}