----- > [!proposition] Proposition. ([[lift uniqueness lemma for covering spaces]]) > Let $p: \widetilde{X} \to X$ be a [[covering space]], $f:Y \to X$ a [[continuous]] map, and $\tilde{f_{0}}, \tilde{f_{1}}: Y \to X$ be two [[lifting|lifts]] of $f$. Then the set $S:= \{ y \in Y : \tilde{f_{0}}(y)=\tilde{f}_{1}(y) \}$ is [[clopen set|clopen]] in $Y$. [[clopen characterization of connectedness|Thus]] if $Y$ is [[connected]] then either $S=\emptyset$ or $S=Y$. ^proposition > [!proof]- Proof. ([[lift uniqueness lemma for covering spaces]]) > **Open.** In general, given any $y \in Y$ (not necessarily in $S$) and obtaining a [[neighborhood]] $X \supset U_{y} \ni f(y)$ [[evenly covered]] by $p$, we have that > Let $s \in S$, so that $\tilde{f}_{0}(s)=\tilde{f}_{1}(s)=: e$. Obtain a [[neighborhood]] $X \supset U \ni f(s)$ [[evenly covered]] by $p$; write $p ^{-1}(U)= \coprod_{\alpha}V_{\alpha}$ for open [[neighborhood]]s $V_{\alpha} \subset \widetilde{X}$ each [[homeomorphism]] to $U$ via $p |_{V_{\alpha}}$. Find the [[neighborhood]] $V_{\beta}$ in which $e$ lives, and consider the open [[neighborhood]] $N:= \tilde{f}_{0}^{-1}(V_{\beta}) \cap \tilde{f}_{1}^{-1}(V_{\beta})$ of $s$. Now, $p |_{V_{\beta}} \circ \tilde{f}_{0} |_{N} =f |_{N}= p |_{V_{\beta}} \circ \tilde{f}_{1}|_{N}$ and since $p |_{V_{\beta}}$ is a [[homeomorphism]] (a [[monomorphism]], in particular), we can conclude $\tilde{f}_{0} |_{N}=\tilde{f}_{1} |_{N}$ and therefore that $y \in N \subset S$. So $S= \text{int }S$. > ![[CleanShot 2024-06-05 at [email protected]|500]] > **Closed.** We'll show $S=\overline{S}$ via [[closure is set together with limit points]]. Let $y \in \overline{S}$ and suppose $\tilde{f}_{0}(y) \neq \tilde{f}_{1}(y)$. This is by definition impossible if $y \in S$, so assume $y$ is a [[limit point]] of $S$. Obtain a [[neighborhood]] $U \subset X$ about $f(y)$; write $p ^{-1}(U)= \coprod_{\alpha}V_{\alpha}$ for open [[neighborhood]]s $V_{\alpha}$ each [[homeomorphism]] to $U$ via $p |_{V_{\alpha}}$. $\tilde{f}_{0}(y)$ lives in one such [[neighborhood]] $V_{\beta_{0}}$ and $\tilde{f}_{1}(y)$ lives in another such [[neighborhood]] $V_{\beta_{1}}$; also $\beta_{0} \neq \beta_{1}$ since $p |_{V_{\beta_{0}}} \circ \tilde{f}_{0}(y) = f(y)= p |_{V_{\beta_{1}}} \circ \tilde{f}_{1}(y)$ and if $\beta_{0}=\beta_{1}=\beta$, then, because $p |_{\beta}$ is a [[bijection]] (in particular a [[monomorphism]]), [[characterization of injectivity and surjectivity|we'd have]] $\tilde{f}_{0}(y)=\tilde{f}_{1}(y)$. Now take the [[neighborhood]] $\tilde{f}_{0}^{-1}(V_{\beta_{0}}) \cap \tilde{f}_{1}^{-1}(V_{\beta_{1}})$ of $y$; since $y$ is a [[limit point]] it intersects $S$ at some point $s \neq y$, $\tilde{f}_{0}(s)=\tilde{f}_{1}(s)$[](characterization%20of%20injectivity%20and%20surjectivity%20in%20Set.md)roof ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch01