Todd Trimble
quotients of reals

Recall that a Peano space is a connected, locally connected, compact metrizable space. The name comes from the fact that these are exactly the spaces that can be filled by curves; cf. Peano’s discovery that there exist curves that fill the space [0,1] 2[0, 1]^2:

Theorem

(Hahn-Mazurkiewicz) For a nonempty Hausdorff space XX, there exists a continuous surjection α:IX\alpha: I \to X from the unit interval I=[0,1]I = [0, 1] if and only if XX is a Peano space.

Perhaps it should be pointed out that these are exactly the (Hausdorff) quotients of II: that if q:IXq: I \to X is a continuous surjection, then qq is in fact a quotient map (simply because qq is a closed surjective map).

We get a wider class of spaces by considering (Hausdorff) quotients of =(,infty)\mathbb{R} = (-\infty, infty) instead.

Remark

Already I’m getting tired of putting “Hausdorff” in parentheses, so let’s agree in this article that we henceforth consider only nonempty Hausdorff spaces.

Note that II itself is a quotient of \mathbb{R}, via the composite

exp(i)S 1I\mathbb{R} \stackrel{\exp(i \cdot -)}{\to} S^1 \to I

where the first map is a covering projection and the second map is a branched projection map. It follows that every Peano space is a quotient of \mathbb{R}.

I will aim to prove the following result:

Theorem(?)

A space XX is a quotient of \mathbb{R} if and only if it is a filtered colimit of Peano spaces.

It should be observed that quotients of locally connected spaces are locally connected.

Lemma

There is a continuous surjection f:Xf: \mathbb{R} \to X if and only if XX is path-connected and covered by countably many Peano subspaces. (N.B.: the topology on XX need not be the coherent topology induced from the union.)

Proof

In the forward direction, given a continuous surjection f:Xf: \mathbb{R} \to X, the space XX is covered by Peano spaces f([n1,n])f([n-1, n]) where nn ranges over \mathbb{Z}. In the backward direction: given Peano subspaces {X n} n\{X_n\}_{n \in \mathbb{Z}} that cover XX, we can construct surjective maps f n:[2n,2n+1]X nf_n: [2n, 2n+1] \to X_n, and then extend the union of these maps to an (evidently surjective) function f:Xf: \mathbb{R} \to X by letting the restriction of ff to the interval [2n1,2n][2n-1, 2n] be a chosen path from f n1(2n1)f_{n-1}(2n-1) to f n(2n)f_n(2n).

Lemma

Every quotient of \mathbb{R} can be expressed as a TopTop-colimit of a countable chain of Peano spaces.

Proof

This is essentially obvious: suppose given a quotient map q:Xq: \mathbb{R} \to X, and let X n=q([n,n])X_n = q([-n, n]) as a subspace of XX. Then the restriction, call it q n:[n,n]X nq_n: [-n, n] \to X_n, is a quotient map, so X nX_n is a Peano space. By compatibility of the restriction maps, the identity function

nX nX\bigcup_n X_n \to X

is a continuous function if we regard the domain as endowed with the coherent or colimit topology (induced by the countable chain of inclusions X nX n+1X_n \hookrightarrow X_{n+1}). This identity function is also open: since qq is a quotient map, the continuity of the inverse id:X nX nid: X \to \bigcup_n X_n is equivalent to the continuity of

q: nX nq: \mathbb{R} \to \bigcup_n X_n

but since the topology of \mathbb{R} is the coherent topology induced from the chain of inclusions [n,n][(n+1),n+1][-n, n] \hookrightarrow [-(n+1), n+1], this boils back down to the continuity of the q nq_n and the compatibility between them.

Lemma

A colimit X X_\infty of a countable chain of Peano spaces f n:X nX n+1f_n: X_n \to X_{n+1} is a quotient of \mathbb{R}.

Proof

Observe that each X n+1X_{n+1} is path-connected. We construct by induction a sequence of quotient maps q n:[n,n]X nq_n: [-n, n] \to X_n such that f nq nf_n \circ q_n is the restriction of q n+1:[(n+1),n+1]X n+1q_{n+1}: [-(n+1), n+1] \to X_{n+1} along the inclusion [n,n][(n+1),n+1][-n, n] \hookrightarrow [-(n+1), n+1]. Indeed, there exists a quotient map r n:[n+12,n+1]X n+1r_n: [n + \frac1{2}, n+1] \to X_{n+1}, and by path-connectedness there exists an extension of the union of the maps

[n,n]f nq nX n+1,[n+12,n+1]r nX n+1[-n, n] \stackrel{f_n \circ q_n}{\to} X_{n+1}, \qquad [n + \frac1{2}, n+1] \stackrel{r_n}{\to} X_{n+1}

to a map q n+1:[(n+1),n+1]X n+1q_{n+1}: [-(n+1), n+1] \to X_{n+1}, evidently a quotient map. The maps paste together to give a surjective map q:X q: \mathbb{R} \to X_\infty. To see qq is a quotient map, suppose q 1(U)q^{-1}(U) is open for some given UX U \subseteq X_\infty. Let i n:X nX i_n: X_n \to X_\infty be the cocone inclusion; then

q n 1i n 1(U)=[n,n]q 1(U)q_n^{-1} i_n^{-1}(U) = [-n, n] \cap q^{-1}(U)

is open in [n,n][-n, n], hence i n 1(U)i_n^{-1}(U) is open in X nX_n since q nq_n is a quotient map, and thus UU is open in X X_\infty since X X_\infty has the coherent topology induced from the i ni_n.

Definition

An LP-space is a colimit in TopTop of a countable chain of Peano spaces.

Lemma

The category of LP-spaces has colimits of countable chains.

Created on May 26, 2014 at 09:15:51 by Todd Trimble