Continuous surjection $\mathbb R^m\to \mathbb R^n$ that is not a quotient map

Question

(I have found examples 1,2 that answer my original questions, so the question here is refined)

This question has been asked many times in this site, but all examples I see are maps between some complicated spaces. So here I'm asking for some examples for good spaces. In fact, it turns out that every continuous function from a path connected space to $\mathbb R$ is a quotient map

Note that the closed map lemma cannot be generalised, for example $(0,1)\to [0,1]$ is not closed. So there might exist continuous surjection from locally compact space to Hausdorff space but not a quotient map, and in the sequel we focus on non-compact manifolds.

Let's first find surjective smooth map between manifolds that is not a quotient map. Such a map must have critical points since every submersion is open(The only obstruction for a smooth surjection not a submersion is that it doesn't have constant rank).

Example 1: For example, $f:(0,2)\to \mathbb S^1,$ where $f(t)=e^{2\pi i\phi(t)}$ where $\phi$ is a smooth bump function that $\phi(0)=0$, increasing in $(0,1)$ and takes value $1$ in $(1,2)$, is not a quotient map, as $(1/2,2)$ is saturated open but its image is not.

If we consider a map onto its image, we also have following example:

Example 2: Consider figure eight $g:(-1,1)\to\mathbb C,g(t)=(1-e^{2\pi i t})\text{sgn} t$. then this map has constant rank, but is not a quotient map onto its image: $(-1,-1/2]$ is closed but its image is not.

By example 2, there is also a smooth map $\mathbb R^n\to \mathbb R^m$ for every $m\geq 2$ of constant rank, viewed as a surjection onto its image, that is not a quotient map. Thus, the result cannot be generalised into arbitary dimensions.

So far, all of these examples have "non-trivial" images. So here I'm asking for nicer maps with very simple image,

(1)Do we have a smooth surjective map $f:\mathbb R^m\to \mathbb R^n$ for every $n>1$ that is not a quotient map.

(2)Do we have a continuous surjective map $f:\mathbb R^m\to \mathbb R^n$ for every $n>1$ that is not a quotient map? For example, is there a spacefilling curve not a quotient map? (Note that we do not always have smooth surjection $\mathbb R^m\to\mathbb R^n$ if $m<n$, but continuous surjection always exists, so (1) doesn't imply (2)).

Answer 1

Here is a construction of surjective continuous maps $f: \mathbb R^m\to \mathbb R^n$ which are not quotient maps, for all $n\ge 2, m\ge 1$. Start with a homeomorphism $h_0: (-\infty, 0]\to [-1,1)$; of course, $h(0)=-1$. Set $J:= (-1,1)\times \{0_{n-1}\}$, an open interval in $\mathbb R^n$ and define $A:= \mathbb R^n\setminus J$. Set $$ p:= (-1,0,...,0)\in \mathbb R^n. $$

The main part will be to find a continuous surjection $f_0: [0,\infty)\to A$ such that $f_0(0)=p$. Once it is found, I will take $f_1: \mathbb R\to \mathbb R^n$ as the map which equals $h_0$ on $(-\infty, 0]$ and equals $f_0$ on $[0,\infty)$. Here I regard $[-1,1)$ as an interval in the 1st coordinate axis in $\mathbb R^n$. Such a map will be clearly not a quotient map since $(-\infty, 0]\cup f_0^{-1}(p)$ is a closed saturated subset of $\mathbb R$ whose image under $f$ is not closed (it is the interval $[-1,1)$). To get a map $f: \mathbb R^m\to \mathbb R^n$ I will then use the composition $$ f_1\circ \pi, $$ where $$ \pi: \mathbb R^m\to \mathbb R $$ is the projection to the first coordinate line.

The existence of $f_0$ is a special case of a more general claim:

Theorem. Suppose that $X$ is a nonempty metrizable space which is an increasing union of countably many compacts $K_i, i\in \mathbb N$, each of which is a Peano continuum (connected and locally connected). Then for every $x_0\in X$ there exists a continuous surjective map $$ g: [0,\infty)\to X, g(0)=x_0. $$ Proof. Without loss of generality, we may assume that $x_0\in K_1$. Then, since $K_1$ is a Peano continuum, there exists a continuous surjective map $g_1=[0,1]\to K_1, g_1(0)=x_0$. I will build surjective maps $g_i: [i-1,i]\to K_i$ inductively, so that $g_{i}(i-1)=g_{i-1}(i-1)$. Then take $g: [0,\infty)\to X$ which equals $g_i$ on $[i-1, i], i\in \mathbb N$. qed

To apply this theorem in the case when $X=A$ and $x_0=p$ as above, observe that $A$ is the increasing union of connected and locally connected compacts $$ K_i=\{x\in \mathbb R^n: 1\le x_1^2 + i x_2^2+...+ i x_n^2\le i^2\}. $$

As for a smooth (actually, real-analytic) example of a surjection $(-1,1)\to S^1$ I would take $$ f(t)= e^{2\pi it^2}. $$

For the existence of dimension-raising proper maps $\mathbb R^m\to \mathbb R^n$, see here.