Characterization of local homeomorphisms

$\def\S{\mathcal{S}} \def\dom{\operatorname{dom}} \def\CS{\mathrm{C}\mathcal{S}} \def\B{\mathcal{B}} \def\im{\operatorname{im}} \def\F{\mathcal{F}} \def\et{\operatorname{\acute Et}}$When one defines sheaves over a topological space $X$ as étale spaces over $X$, usually one first constructs a set $E$ along with a projection $p:E\to X$, and then topologizes $E$ in terms of a certain family of sections of $p$, verifying after that $p$ becomes a local homeomorphism.

This is usually explained to the mathematical reader in an informal way, leaving the details of the construction to them. This post aims to fill up those missing details through the development of the mini-theory of "section bases" (Definition 2), which allow to characterize local homeomorphisms (Proposition 4). The last result, Corollary 9, explains then how to realize in general the construction referred on the first paragraph.

Definition 1. Let $E$, $B$ be spaces and let $p:E\to B$ be a function (not necessarily continuous).

1. A local section of $p$ at $b\in B$ is a continuous map $s:U\to E$, where $U\subset B$ is an open neighborhood of $b$, such that $p\circ s=1_U$.
2. Given a family of local sections $\S$ of $p$, we say that $\S$ covers $E$ if the union of the images of the sections of $\S$ equals $E$, $$ E=\bigcup_{s\in\S}\im s. $$

Given spaces $E$ and $B$ and a function $p: E\to B$, we denote $\S_p$ the family of local sections of $p$. Given a family $\S\subset\S_p$, we denote $\dom\S=\{\dom s\mid s\in\S\}$ to the set of domains of the sections in $\S$ (which is a set of open sets of $B$), and $\im\S=\{\im s\mid s\in\S\}$ to the set of images. In particular, $\S$ covers $E$ iff $\im\S$ is a cover of $E$ (not necessarily open). And if $\S$ covers $E$, then $p(E)$ is open in $B$ and $\dom\S$ must be an open cover of $p(E)$. We denote $\S_p^\mathrm{cont}\subset \S_p$ to the family of continuous local sections of $p$.

Definition 2. Let $E$, $B$ be spaces and let $p:E\to B$ be a function (not necessarily continuous).

• A section basis of $p$ is a family $\S\subset\S_p$ with the properties:

1. $\S$ covers $E$.
2. (Shrinking property on the base space) For each $s\in\S$ and each $b\in\dom s$, the set $N_s(b)=\{U\subset\dom s: U\text{ open, }b\in U, s|_U\in\S\}$ is a neighborhood system of $b$, meaning that for every open neighborhood $V\subset B$ of $b$, there is $U\in N_s(b)$ with $U\subset V$.
3. (Shrinking property on the total space) For all $s_1,s_2\in\S$ and all $e\in s_1(U_1)\cap s_2(U_2)$, where $U_i=\dom s_i$, there is $s\in\S$ such that $p(e)\in U=\dom s\subset U_1\cap U_2$ and $s_1|_U=s=s_2|_U$.

• A continuous section basis of $p$ is a section basis of $p$ made up of continuous functions.
  

Remarks 3. Let $\S$ be a section basis of $p$.

• This implies that $\im S$ is an abstract basis of the underlying set of $E$, and the topology generated by $\im S$ may or may not coincide with that of $E$ (they may not even be commensurable). We say that the section basis $\S$ generates $E$ whenever the (set) basis $\im\S$ generates the topological space $E$.
• The set $\dom\S$ is a basis that generates the topological space $p(E)$.

Proposition 4. Let $E$, $B$ be spaces and let $p:E\to B$ be any function. The following are equivalent:

$(1)$ $p$ is a local homeomorphism.

$(2)$ $\S_p^\mathrm{cont}$ covers $E$ and all continuous local sections of $p$ are open maps (equivalently, open embeddings).

$(2')$ There is a family of continuous sections $\S\subset\S_p^\mathrm{cont}$ that covers $E$ and such that for all $s\in\S$, the section $s$ is an open map (equivalently, an open embedding).

$(3)$ $E$ has the final topology with respect to any continuous section basis of $p$, and $(3')$ holds.

$(3')$ There is a section basis $\S$ of $p$ such that $E$ has the final topology with respect to this basis. (It follows that $\S$ is a continuous section basis.)

$(4)$ All continuous section bases of $p$ generate the topological space $E$, and $(4')$ holds.

$(4')$ There is a section basis $\S$ of $p$ that generates the topological space $E$. (It follows that $\S$ is a continuous section basis.)

Moreover, if any of these conditions hold, then $\S_p^\mathrm{cont}$ is a section basis of $p$.

Remarks 5.

• We do not require $p$ to be continuous. This follows from any of the stated equivalent properties.
• For each $n=2,3,4$, the properties $(n),(n')$ characterize local homeomorphisms through their local sections in a different direction. Namely, $n=2$ does it in terms of openness of l.s., $n=3$ in terms of finality of $E$ with respect to the l.s., and $n=4$ in terms of section bases.
• We have $(n)\Rightarrow(n')$ automatically, for $n=2,3,4$.

Lemma 6. Let $p:E\to B$ be a local homeomorphism, and let $s:V\to E$ and $s':V'\to E$ be continuous local sections of $p$. Suppose $b\in V\cap V'$. Then $s_b=s_b'$ if and only if $s(b)=s'(b)$.

Proof. Let $s:V\to E$ be a local section of $p$, and let $b_0\in V$. Let $U\subset E$ be an open neighborhood of $s(b_0)$ such that $p|_U$ is an open embedding. Denote $f:p(U)\to U$ to the continuous inverse of $p|_U:U\to p(U)$. Then $p(U)\cap s^{-1}(U)$ is a non-empty open set, since it contains $b_0$. We will prove the identity $f|_{p(U)\cap s^{-1}(U)}=s|_{p(U)\cap s^{-1}(U)}$. Let $b\in p(U)\cap s^{-1}(U)$. We have $s(b)\in U$ and there exists a unique $e\in U$ such that $p(e)=b$. Thus $s(b)=e$, and $f(b)=f(p(e))=e=s(b)$.

This shows that the germ of $s$ at $b_0\in V$ is determined by the germ of a local inverse of $p$ at $s(b_0)$. Thus, we obtain the claim. $\square$

Proof of Proposition 4.

$(1)\Rightarrow(2)$ It is clear that $\S_p^\mathrm{cont}$ covers $E$, since $p$ has continuous local inverses. We also have that continuous local sections are open: a continuous local section $s:V\subset B\to E$ of $p$ is locally open since, by last lemma, its germ at a point $b\in V$ coincides with the germ of a bicontinuous local inverse of $p$ at $s(b)$. Therefore $s$ is locally open and therefore open.

$(2)\Rightarrow (3)$ Any topological space is final with respect to any open cover. Thus, any topological space is final with respect to a family of open immersions into the space that covers it. We show that also $(3')$ holds showing that $\S_p^\mathrm{cont}$ is a section basis. It covers $E$ and it satisfies the shrinking property on the base space. To see that it also satisfies shrinking on the total space, let $s_i:U_i\subset B\to E$ be local sections of $p$, for $i=1,2$. Then $V=s_1(U_1)\cap s_2(U_2)$ is open since $s_i$ is open. Thus $s_i^{-1}(V)=p(V)$ is open, and $s=s_1|_{p(V)}=s_2|_{p(V)}$ is a continuous local section of $p$ such that $\im s=V$.

$(3)\Rightarrow (4)$ Let $\S\subset\S_p^\mathrm{cont}$ be a section basis of $p$ such that $E$ has the final topology with respect to $\S$. We show that $\im \S$ generates $E$. Let $s:U\to E$ be a local section inside $\S$. Seeing that $s(U)$ is open amounts to seeing that $t^{-1}(s(U))$ is open, for $t:V\to E$ inside $\S$. Let $b_0\in t^{-1}(s(U))$. Then $t(b_0)\in t(V)\cap s(U)$. By the shrinking property on the total space, there is a local section $g:W\to E$ such that $b_0=p(t(b_0))\in W\subset V\cap U$ and $g=t|_W=s|_W$. Thus $b_0\in W\subset t^{-1}(s(U))$, since for $b\in W$, we have $t(b)=s(b)\in s(U)$. Thus $t^{-1}(s(U))$ is open in $V$.

It is left to check that open sets of $E$ are unions of open sets of $\im\S$. Let $G\subset E$ be open, $e\in G$, and pick $s:U\to E$ in $\S$ such that $e\in s(U)$. Then $s^{-1}(G)$ is an open neighborhood of $p(e)$ and by the shrinking property on the base space, there is $U'\subset U$ such that $p(e)\in U'\subset s^{-1}(G)$ and $s|_{U'}\in\S$. Thus $e\in\im s|_{U'}\subset G$.

Finally, $(4')$ also holds since $(3')$ holds and so does the first part of $(3)$.

$(4')\Rightarrow (2')$ Let $\S\subset\S_p$ be a section basis of $p$ generating $E$, and let $s:U\to E$ be in $\S$. Then $s$ is open, since it carries the elements of the basis $N=\bigcup_{b\in U}N_s(b)$ of $U$ to open sets in $E$. (The notation $N_s(b)$ is from the definition of section basis.) Showing that $\S$ is a continuous section basis amounts to showing that $t^{-1}(s(U))$ is open for all $s,t\in\S$, where $U=\dom s$. This is done as in the first paragraph of the proof of $(3)\Rightarrow (4)$.

$(2')\Rightarrow (1)$ Let $e\in E$ and let $s:U\to E$ be a local section from $\S$ such that $s(U)\ni e$. Then $p|_{s(U)}:s(U)\to U$ and $s:U\to s(U)$ are inverse functions and the latter is a homeomorphism, hence, so is the former.

So far we have proven the implications depicted in this diagram:

It is left to show that $(3')$ implies any of the rest.

$(3')\Rightarrow (2')$ Let $\S\subset\S_p$ be a section basis such that $E$ has the final topology with respect to $\S$. We show that $s:U\to E$ in $\S$ is open. For that, since $N_s(b)$ is a basis of $U$, it suffices to show that $s(U)$ is open. This amounts to showing that $t^{-1}(s(U))$ is open for all $t\in\S$, and this can be done in the same manner as it was done in proof of $(3)\Rightarrow (4)$. $\square$

There are times in which one begins with a map $p:A\to B$, where $B$ is a space and $A$ is just a set, and one wishes to topologize $A$ in terms of “local sections” of $p$. We explain how this can be done.

Definition 7. Let $p:A\to B$ be a function, where $B$ is a space and $A$ is a set. We define the terms local section of $p$ and family of local sections of $p$ covering $A$ in the formally same way as in Definition 1, and we define an abstract section basis of $p$ to be a collection $\S$ of local sections of $p$ that satisfies the formally identical properties of Definition 2.

If $\S$ is an abstract section basis of $p:A\to B$, then $\im\S$ is an abstract set basis of $A$. We will call the topology generated by $\S$ on $A$ to the topology generated by $\im\S$ on $A$. We will now relate the concept of abstract section basis to the previous results.

Lemma 8. Let $p:A\to B$ be a function from a set $A$ to a space $B$, and let $\S$ be an abstract local basis of $p$. Then, with respect to the topology on $A$ generated by $\S$, the sections of $\S$ are open embeddings.

Proof. Sections are always injective, and they are open by the same argument that in $(4')\Rightarrow(2')$ of Proposition 4. Lastly, to show continuity of $t\in\S$, it suffices to show that if $s:U\to A$ is in $\S$, then $t^{-1}(s(U))$ is open. This is done in the same manner as in $(3)\Rightarrow(4)$ of Proposition 4. $\square$

Corollary 9. Let $p:A\to B$ be a map from a set $A$ to a space $B$, and let $\S$ be an abstract section basis of $p$. Then, with respect to the topology on $A$ generated by $\S$, we have that $\S$ is a continuous section basis of $p$ and $p$ is a local homeomorphism.

Proof. By last lemma, sections of $\S$ are continuous and also point $(2')$ of Proposition 4 holds. $\square$

Example 10. The following is the canonical example of an abstract section basis: Let $X$ be a topological space, and let $\F:\operatorname{Ouv}(X)^\mathrm{op}\to\mathsf{Set}$ be a presheaf of sets over $X$. Define $\et(\F)=\bigsqcup_{x\in X}\F_x$. We have a map $p:\et(\F)\to X$ that sends each element of $\F_x$ to $x$. For each open $U\subset X$ and each $s\in\F(U)$, denote

$$\begin{align*} \sigma_s:U&\to\et(\F)\\ x&\mapsto s_x. \end{align*}$$

Let $\mathcal{B}$ be a basis of $X$. Then the set $\S=\{\sigma_s\mid s\in\F(B),\;B\in\mathcal{B}\}$ is a section basis of $p$ (check!). This way, by Corollary 9, $p:\et(\F)\to X$ becomes an étale space over $X$. This is why $\et(\F)$ is called the étale space of the presheaf $\F$.