I read a proof about classification of connected one dimensional manifolds.It uses Zorn Lemma to construct a maximal parametrization by arclength and shows this parametrization is global and thus diffeomorphism.
I wonder why this seemingly intuitive theorem requires Zorn lemma.If we cannot use Zorn lemma,is there any counterexample?
It depends what you mean by manifold. If you only assume your space is Hausdorff and locally homeomorphic to $\mathbb R,$ then the long line is a counterexample in ZFC. It can even be given the structure of a differentiable manifold.
For second-countable manifolds, you don’t need any choice - the countable basis gives you all the choices you need.
For paracompact manifolds there are counterexamples. While you can parameterize any compact subset, without countable choice you cannot piece these together. The space will be a differentiable manifold satisfying:
We can use a model of ZFA (ZF with atoms) called a permutation model. The idea is that the operations of ZFA cannot let you break out of a collection of symmetries. Certain paradoxical properties of objects, such as the space below, can be transferred to a model of ZF using the Jech-Sochor embedding theorem.
Let $A$ be a copy of the real line. Let $\mathcal F$ be the set of closed subsets $F\subset A$ of finite measure. Let $G$ be the group of bilipschitz orientation-preserving diffeomorphisms of $A,$ and for $F\in\mathcal F$ let $G_F$ denote the subgroup of diffeomorphisms that fix each $x\in F.$ The collection of groups $\{G_F\}$ are a normal filter base. This specifies a permutation model $N.$
$N$ contains the set $A.$ It has a differentiable structure, with the usual atlases for bounded sets. But there is no injection from $A$ to $\mathbb R$ in this model. Given any $F\in\mathcal F,$ we can pick a $\pi\in G_F$ that moves an element $a\not\in F,$ so any function $f:A\to\mathbb R$ fixed by $G_F$ has to satisfy $f(a)=(\pi f)(\pi a)=f(\pi a).$
To verify property 1, work outside the permutation model for now, so we can use choice and identify $A$ with the real line. We are given an open cover $\mathcal U.$ Let $\mathcal U_1$ be the collection of open intervals $(a,b)$ of length at most one such that there exists $U\in\mathcal U$ with $(a,b)\subset U.$ For each $n\in\mathbb Z$ pick a finite set $\mathcal V_n\subset\mathcal U_1$ covering $[n,n+1].$ The union $\mathcal V=\bigcup\mathcal V_n$ is a locally finite cover of $A.$ Let $F’$ be the discrete set of endpoints of intervals in $\mathcal V.$ The set $\mathcal V$ is fixed by $G_{F’},$ and we can self-index this collection by defining $J=\mathcal V$ and $V_j=j$ for $j\in J.$ This verifies property 1.
For property 2, again working externally, apply the argument in the previous paragraph and let $x_n$ be an enumeration of the set $F’$ that was constructed in that argument. Take $F’’=\bigcup_n [x_n-2^{-n},x_n+2^{-n}].$ For each $j=(x_n,x_m)$ in $J,$ pick a smooth function $\sigma_j$ that equals zero outside $j,$ is strictly positive in the interior of $j,$ and is constant in the interval $[x_n+2^{-n},x_m-2^{-m}].$ The last condition is vacuously true if $x_n+2^{-n}\geq x_m-2^{-m}.$ Define $\rho_j(x)=\sigma_j(x)/\sum_{k\in J} \sigma_k(x).$ Then $\{\rho_j:j\in\mathcal J\}$ is a partition of unity subordinate to $\mathcal U$ with compact supports, and is fixed by $G_{F’’}.$