Let $p : Y \to X$ be a covering projection and $U \subset X$ be open. An open subset $V \subset p^{-1}(U)$ is called plain over $U$ if the restriction $p_V : V \to U$ of $p$ is a homeomorphism. A sheet structure over an open $U \subset X$ is a set $S(U)$ of pairwise disjoint open subsets of $p^{-1}(U)$, called sheets, such that
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$\bigcup_{V \in S(U)} V = p^{-1}(U)$
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Each $V \in S(U)$ is plain over $U$.
A nonempty open subset $U \subset X$ is called evenly covered if there exists a sheet structure over $U$. The sheet number of an evenly covered $U$ is the cardinality of a sheet structure over $U$ (which is the same for all sheet structure over $U$), or equivalently, the common cardinality of the fibers $p^{-1}(x)$ with $x \in U$.
What can be said about sheet structures over $U$? In particular, when are they unique?
The following facts are scattered through the literature and this forum or belong to ''mathematical folklore'':
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Let $U =\bigcup_{\alpha \in A} U_\alpha$ with pairwise disjoint open evenly covered $U_\alpha \subset X$ having sheet structures $S(U_\alpha)$ with the same cardinality $\mathfrak c$. Then $U$ is evenly covered. More precisely, let $\phi_\alpha : C \to S(U_\alpha)$ be bijections defined on an index set $C$ of cardinality $\mathfrak c$, then $S(U,\phi_\alpha) = \{ \bigcup_{\alpha \in A} V_{\phi_\alpha(c)} \mid c \in C \}$ is a sheet structure over $U$. Each of the sheets in this structure contains exactly one sheet of each $S(U_\alpha)$. If the cardinalities of $C$ and $A$ are $> 1$, then there exist quite a number of distinct sheet structures over $U$. In fact, in that case $S(U,\phi_\alpha) = S(U,\phi'_\alpha)$ if and only if all $\phi_\alpha = \phi'_\alpha$.
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Let $U$ be evenly covered with a sheet structure $S(U)$. For each open $U' \subset U$ and each $V \in S(U)$ define $V \mid_{U'} = V \cap p^{-1}(U') = p_V^{-1}(U') \subset V$. Then the restriction $S(U) \mid_{U'} = \{V \mid_{U'} \mid V \in S(U) \}$ is a sheet structure over $U'$. The restriction map $\rho : S(U) \to S(U) \mid_{U'}, \rho(V) = V \mid_{U'}$, is a bijection. Moreover, for each open $V' \subset V \in S(U)$ one gets an open $U' = p_V(V') = p(V) \subset U$ and one has $V' = V \mid_{U'} \in S(U) \mid_{U'}$.
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Let $U$ be evenly covered with sheet number $1$. Then there trivially exists a unique sheet structure over $U$.
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Let $U$ be evenly covered with sheet number $> 1$. Then there exists a unique sheet structure over $U$ if and only if $U$ is connected. If $U$ is connected, then the unique sheet structure over $U$ consists of the connected components of $p^{-1}(U)$.
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If $U$ is an evenly covered set, then for each open $W \subset p^{-1}(U)$ which is plain over $U$ there exists a sheet structure over $U$ which contains $W$.
The question is to give proofs of these facts to obtain a standard reference in this forum.
Best Answer
Points 1. - 3. are trivial.
Proof of 4. :
If $U$ is not connected, then we have a decomposition $U = U_1 \cup U_2$ with nonempty disjoint open $U_i$. Let $S(U)$ be a sheet structure over $U$. Then we obtain sheet structures $S(U_i) = S(U) \mid_{U_i}$ over $U_i$. Now 1. applies to give distinct sheet structures over $U$. Conversely, let $U$ be connected. Let $S(U)$ be a sheet structure over $U$. Then each $V \in S(U)$ is connected, and moreover $V$ is a maximal connected subset of $p^{-1}(U)$ because any bigger connected $C \supset V$ would meet the open set $V^* = \bigcup_{V' \in S(U), V' \ne V} V'$. This would split $C$ into the disjoint nonempty open subsets $V$ and $C \cap V^*$. Thus each $V \in S(U)$ is a component of $p^{-1}(U)$. Since $\bigcup_{V \in S(U)} V = p^{-1}(U)$, we conclude that each component of $p^{-1}(U)$ is contained in $S(U)$.
Proof of 5. :
Let $S(U)$ be a sheet structure over $U$. Then all $W_V = W \cap V$, $V \in S(U)$, are open subsets of $W$ (some may be empty) which cover $W$ and are mapped by the homeomorphism $p_W : W \to U$ onto open $U_V \subset U$. Note that $p_W(W \cap V) = p_V(W \cap V)$. The $U_V$ cover $U$ and are pairwise disjoint because the $W_V$ are pairwise disjoint. Let $A = \{ V \in S(U) \mid W_V \ne \emptyset \}$. All $U_V$ with $V \in A$ are evenly covered with sheets structures $S(U) \mid_{U_V}$. For each $V \in A$ the restriction map $\rho_V : S(U) \to S(U) \mid_{U_V}$ is a bijection such that $\rho_V(V) = p_V^{-1}(U_V) = p_W^{-1}(U_V) = W_V$. Now fix $V_0 \in A$ and let $\phi_V : S(U) \to S(U) \mid_{U_V}$ be the bijection agreeing with $\rho_V$ for $V' \ne V,V_0$ and satisfying $\phi_V(V_0) = V, \phi_V(V) = V_0$. Then $W = \cup_{V \in A} \phi_V(V_0) \in S(U,\phi_V)$.