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 $[0, 1]^2$ :

Theorem

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

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

We get a wider class of spaces by considering (Hausdorff) quotients of $\mathbb{R} = (-\infty, infty)$ 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 $I$ itself is a quotient of $\mathbb{R}$ , via the composite

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

Proof

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

Lemma

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

Proof

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

\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 $X_n \hookrightarrow X_{n+1}$ ). This identity function is also open: since $q$ is a quotient map, the continuity of the inverse $id: X \to \bigcup_n X_n$ is equivalent to the continuity of

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

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

Lemma

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

Proof

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

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

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

is open in $[-n, n]$ , hence $i_n^{-1}(U)$ is open in $X_n$ since $q_n$ is a quotient map, and thus $U$ is open in $X_\infty$ since $X_\infty$ has the coherent topology induced from the $i_n$ .

Definition

An LP-space is a colimit in $Top$ 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