Nakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
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已登记的原文定义
Nakahara · second-edition-local · §4.6.5 · PDF 第168页
Definition 4.12. Let $X$ and $\widetilde{X}$ be connected topological spaces. The pair $(\widetilde{X}, p)$ , or simply $\widetilde{X}$ , is called the covering space of $X$ if there exists a continuous map $p: \widetilde{X} \to X$ such thatNakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §4.6.5 · PDF 第168页
(1) $p$ is surjective (onto)
(2) for each $x \in X$ , there exists a connected open set $U \subset X$ containing $x$ , such that $p^{-1}(U)$ is a disjoint union of open sets in $\widetilde{X}$ , each of which is mapped homeomorphically onto $U$ by $p$ .Nakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §4.6.5 · PDF 第168页
In particular, if $\widetilde{X}$ is simply connected, $(\widetilde{X}, p)$ is called the universal covering space of $X$ . [Remarks: Certain groups are known to be topological spaces. They are called topological groups. For example $\mathrm{SO}(n)$ and $\mathrm{SU}(n)$ are topological groups. If $X$ and $\widetilde{X}$ in definition 4.12 happen to be topological groups and $p : \widetilde{X} \to X$ to be a group homomorphism, the (universal) covering space is called the (universal) covering group.]Nakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §4.6.5 · PDF 第168页
For example, $\mathbb{R}$ is the universal covering space of $S^1$ , see section 4.3. Since $S^1$ is identified with $\mathrm{U}(1)$ , $\mathbb{R}$ is a universal covering group of $\mathrm{U}(1)$ if $\mathbb{R}$ is regarded as an additive group. The map $p: \mathbb{R} \to \mathrm{U}(1)$ may be $p: x \to \mathrm{e}^{\mathrm{i}2\pi x}$ . Clearly $p$ is surjective and if $U = \{\mathrm{e}^{\mathrm{i}2\pi x} | x \in (x_0 - 0.1, x_0 + 0.1)\}$ , thenNakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §4.6.5 · PDF 第168页
$$
p ^ {- 1} (U) = \bigcup_ {n \in \mathbb {Z}} (x _ {0} - 0. 1 + n, x _ {0} + 0. 1 + n)
$$
Nakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
Nakahara · second-edition-local · §4.6.5 · PDF 第168页
which is a disjoint union of open sets of $\mathbb{R}$ . It is easy to show that $p$ is also a homomorphism with respect to addition in $\mathbb{R}$ and multiplication in $\mathrm{U}(1)$ . Hence, $(\mathbb{R}, p)$ is the universal covering group of $\mathrm{U}(1) = S^1$ .Nakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。
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Nakahara · second-edition-local
已登记出处中的最早位置：§4.6 · zh-CN。
原文中已定位的引用（不自动视为依赖）
- href=“06-section-4-6.md#definition-4-12”
- href=“02-section-4-2.md#bound-definition-dd3051b34cdd1c57cb935a6d”
- [注记：某些群已知是拓扑空间，它们称为<a class=“concept-reference” href=”../../../../../concepts/by-id/278b7ab5-c1c7-44b6-9b34-05f394a2a1cb.md”><u class=“concept-term” id=“bound-first-acb5acfdbbcbe52713d54e77” data-concept-identity=“urn:uuid:278b7ab5-c1c7-44b6-9b34-05f394a2a1cb” data-concept-role=“first-mention” data-concept-first-mention=“true” data-concept-navigation-schema=“1” data-concept-source-node=“a-f93e176dae385677718561c6” data-concept-identity-status=“formal” data-concept-definition-status=“mention” data-concept-route=“/concepts/by-id/278b7ab5-c1c7-44b6-9b34-05f394a2a1cb” title=“查看正式概念页”>拓扑群(topological group)</u></a>。例如 与 都是拓扑群。若 [定义 4.12](06-section-4-6.md#definition-4-12)
- [4.3 节](03-section-4-3.md#section-4-3)
- href=“06-section-4-6.md#theorem-4-10”
- href=“06-section-4-6.md#example-4-12”