光滑流形预备

• 前置
• 定义
• 性质

前置

• 连续映射

定义

• If $\forall t\in T,X_t$ is a set, their disjoint union is the set ${⨆}_{t\in T}{X}_t=\{(x,t)\mid x\in X_t,t\in T\}$
• Given an indexed family of topological spaces $\forall t\in T,X_t$ is a topological space, we define the disjoint union topology on ${⨆}_{t\in T}{X}_t$ by declaring a subset of ${⨆}_{t\in T}{X}_t$ to be open if and only if its intersection with each $X_t$ is open in $X_t$.

性质

1. Let $X$ be a second-countable topological space. Every open cover of $X$ has a countable subcover.
2. CHARACTERISTIC P ROPERTY: If $\forall t\in T,(X_t,{\tau }_t)$ are topological spaces, a map $F:B\to {\prod }_{t\in T}{X}_t$ is continuous if and only if each of its component functions $F_t={\pi }_t\circ F:B\to X_t$ is continuous.
3. The product topology is the unique topology on ${\prod }_{t\in T}{X}_t$ for which the characteristic property holds
4. Given any continuous maps $F_t:X_t\to Y_t$ for $t\in T$, the product map
   $\mathbf{F}:{\prod }_{t\in T}{X}_t\to {\prod }_{t\in T}{Y}_t$ is continuous, where ${\mathbf{F}}_t(x)=F_t(x_t)$ for all $x\in {\prod }_{t\in T}{X}_t$
5. If $S_t$ is a subspace of $X_t$ for $t\in T$, the product topology and the subspace topology on ${\prod }_{t\in T}{S}_t\subseteq {\prod }_{t\in T}{X}_t$ coincide.
6. For any $t\in T$ and any choices of points $a_j\in X_j$ for $j\ne i$, the map $x\mapsto (x_t,t\in T|x_i=x\wedge x_j=a_j)$ is a topological embedding of $X_i$ into the product space $\prod X_1$.
7. Any product of Hausdorff spaces is Hausdorff.
8. Countable product of first-countable spaces is first-countable.
9. Countable product of second-countable spaces is second-countable.
10. (Properties of the Disjoint Union Topology). Suppose $\forall t\in T,X_t$ is a topological space, and ${\bigcup }_{t\in T}{X}_t$ is endowed with the disjoint union topology.
11. CHARACTERISTIC P ROPERTY: If $Y$ is a topological space, a map $F:{⨆}_{t\in T}{X}_t\to Y$ is continuous if and only if $F\circ {\iota }_t:X_t\to Y$ is continuous for each $t\in T$. where ${\iota }_t:X_t\to {⨆}_{t\in T}{X}_t:x\mapsto (x,t)$
12. The disjoint union topology is the unique topology on ${⨆}_{t\in T}{X}_t$ for which the characteristic property holds.
13. A subset of ${⨆}_{t\in T}{X}_t$ is closed if and only if its intersection with each $X_t$ is closed.