nLab long line




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



The ordinary “line\mathbb{R} (i.e. the real numbers equipped with their Euclidean metric topology) may be thought of as the result of gluing a countable set of copies of the half-open interval [0,1)[0, 1) end-to-end in both directions. A long line is similarly obtained by gluing an uncountable set of copies of [0,1)[0, 1) end-to-end in both directions, and is demonstrably “longer” than an ordinary line on account of a number of peculiar properties.

The long line is a source of many counterexamples in topology.



Let ω 1\omega_1 be the first uncountable ordinal, and consider the half-open interval [0,1)[0, 1) as an totally ordered set. A long ray is the ordered set ω 1×[0,1)\omega_1 \times [0, 1) taken in the lexicographic order; as a space, it is given the order topology. The long line is the space obtained by gluing two long rays together at their initial points.

The long line is a line in the sense of being a 11-dimensional locally Euclidean space (without boundary) that is not closed (so not a circle). However, it is not paracompact, so it is not homeomorphic to the real line (even though it is Hausdorff).


All the assertions below apply to both the long line and the long ray. We write LL to cover both cases even if we only treat one.

  1. Every continuous function f:Lf\colon L \to \mathbb{R} is eventually constant, i.e., there exists xLx \in L and cc \in \mathbb{R} such that f(y)=cf(y) = c whenever yxy \geq x (and similarly ff is constant for all sufficiently small xx).

  2. LL is a normal (T 4T_4) space, but the Tychonoff product L×L¯L \times \bar{L} with its one-point compactification is not normal. (See for example Munkres.)

  3. Every continuous map LLL \to L has a fixed point.

  4. LL is sequentially compact but not compact. (Being sequentially compact, they are also countably compact spaces.) Thus, the image f(L)f(L) under a continuous map f:Lf: L \to \mathbb{R}, being sequentially compact, is a compact subspace of \mathbb{R}.

  5. The long ray/line is not contractible. Proof sketch for the long ray: Suppose H:I×LLH \colon I \times L \to L is a homotopy such that H(0,)H(0, -) is constant and H(1,)H(1, -) is the identity. For each t[0,1]t \in [0, 1] the image imH(t,)\im H(t, -) is an interval (either bounded or unbounded), since LL is connected. One may show the set

    S={tI:imH(t,)is bounded}S = \{t \in I: \im H(t, -)\; \text{is bounded}\}

    is both closed and open. It also contains 00, hence is all of II. But SS can’t contain t=1t = 1, contradiction.

Let us flesh out this sketched proof. First we show SS is closed. Denote the bottom element of LL by \bot. If t nt_n is a sequence in SS with limit point tIt \in I, then H({t n}×L)[,b n]H(\{t_n\} \times L) \subseteq [\bot, b_n] for some sequence of bounds b nLb_n \in L; this sequence has an upper bound bLb \in L. Then H({t n}×L)[,b]H(\{t_n\} \times L) \subseteq [\bot, b] for all nn \in \mathbb{N}, which is the same as saying n{t n}×LH 1([,b])\bigcup_n \{t_n\} \times L \subseteq H^{-1}([\bot, b]). Now {t}×L\{t\} \times L is included within the closure of the union, which is included within the closed set H 1([,b])H^{-1}([\bot, b]). But {t}×LH 1([,b])\{t\} \times L \subseteq H^{-1}([\bot, b]) means H({t}×L)[,b]H(\{t\} \times L) \subseteq [\bot, b]; hence tSt \in S.

Now we show SS is open. This uses a countably compact version of the tube lemma. If tSt \in S, then we can find bLb \in L such that {t}×LH 1([,b))\{t\} \times L \subseteq H^{-1}([\bot, b)), where the right side is open in I×LI \times L. Let B r(t)IB_r(t) \subseteq I denote the open ball of radius rr centered at tt, and for c<dc \lt d in LL let (c,d)(c, d) denote the open interval between cc and dd. Consider the collection TT of all triples (n,c,d)×L×L(n, c, d) \in \mathbb{N} \times L \times L such that

U n,c,dB 1/n(t)×(c,d)H 1([,b)).U_{n, c, d} \coloneqq B_{1/n}(t) \times (c, d) \subseteq H^{-1}([\bot, b)).

For each fixed nn \in \mathbb{N}, put U n= (n,c,d)T(c,d)U_n = \bigcup_{(n, c, d) \in T} (c, d). The U nU_n form a countable open cover of LL, since for every xLx \in L there is a U n,c,dU_{n, c, d} containing the pair (t,x)(t, x). Since LL is countably compact, there is a finite subcover U n 1,,U n kU_{n_1}, \ldots, U_{n_k}. Putting n=max{n 1,,n k}n = \max \{n_1, \ldots, n_k\}, we see

B 1/n(t)×L=B 1/n(t)× i=1 kU n i= i=1 kB 1/n(t)×U n i i=1 kB 1/n i(t)×U n iH 1([,b))B_{1/n}(t) \times L = B_{1/n}(t) \times \bigcup_{i=1}^k U_{n_i} = \bigcup_{i=1}^k B_{1/n}(t) \times U_{n_i} \subseteq \bigcup_{i=1}^k B_{1/n_i}(t) \times U_{n_i} \subseteq H^{-1}([\bot, b))

from which it immediately follows that B 1/n(t)SB_{1/n}(t) \subseteq S.


  • Steen and Seebach, Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995.

  • James Munkres, Topology (2nd edition). Prentice-Hall, 2000.

See also

Last revised on July 28, 2018 at 23:47:39. See the history of this page for a list of all contributions to it.