1. Topology

1.1Topological Space

定义 1.1.1. A topological space $(X,\mathcal{O})$ is a set together with open sets $\mathcal{O}\subset \mathcal{P}(X)$, s.t.

(1)	$\varnothing$, $X\in \mathcal{O}$;
(2)	$U_1$, $U_2\in \mathcal{O}$ $\Rightarrow$ $U_1\cap U_2\in \mathcal{O}$;
(3)	$U_i\in \mathcal{O}$, $i\in I$ $\Rightarrow$ $\bigcup _{i\in I}{U}_i\in \mathcal{O}$.

例 1.1.2.  

(1)	$\mathcal{O}=\{\varnothing ,X\}$ the trivial topology on $X$.
(2)	$\mathcal{O}=\mathcal{P}(X)$ the discrete topology.
(3)	the metric topology on a metric space.

1.2Metric Spaces

定义 1.2.1. A metric space $(X,d)$ is a set $X$ together with $$\begin{array}{rlr}d:X\times X& \to \mathbb{R}(x,y)& \mapsto d(x,y)\end{array}$$s.t.

(1)	$d(x,y)⩾0$ with “$=$” if and only if $x=y$;
(2)	$d(x,y)=d(y,x)$;
(3)	$d(x,z)⩽d(x,y)+d(y,z)$, $\forall x,y,z\in X$.

In the metric topology, a subset $U\subset X$ is open if $\forall x\in U$, $\exists \epsilon >0$, s.t.$$B(x,\epsilon ):=\{y\in X\mid d(x,y)<\epsilon \}\subset U.$$

Terminology. Let $(X,\mathcal{O})$ be a topological space.

(1)	$V\subset X$ is closed if $X\setminus V\in \mathcal{O}$.
(2)	$x\in X$, $W\subset X$ is a neighborhood of $x$ in $(X,\mathcal{O})$, if $x\in W$ and $W$ contains an open set $U$, s.t. $x\in U\subset W$.
(3)	$U_i\in \mathcal{O}$, $i\in I$, the $U_i$ form an open cover of $X$ if ${\bigcup }_{i\in I}{U}_i=X$.

定义 1.2.2. A topological space $(X,\mathcal{O})$ is Hausdorff if $\forall x_1,x_2\in X$, $x_1\ne x_2$, $\exists U_1,U_2\in \mathcal{O}$, s.t. $x_i\in U_i$ and $U_1\cap U_2=\varnothing$.

例 1.2.3. The metric topology of a metric space is always Hausdorff.

证明. Let $x,y\in X$ and $x\ne y$. Then $d(x,y)>0$. Take $\epsilon :=\frac{d(x,y)}{2}$, then $B(x,\epsilon )\cap B(y,\epsilon )=\varnothing$ and $x\in B(x,\epsilon )$, $y\in B(y,\epsilon )$.

$\square$

1.3Basis of Topology

定义 1.3.1. A basis of the topology $\mathcal{O}$ is a $B\subset \mathcal{P}(X)$, s.t. every $U\in \mathcal{O}$ is a union of subsets in $B$.

引理 1.3.2. Consider ${\mathbb{R}}^n$ with the metric topology induced by Euclidean distance function $$d(x,y)={\left(\sum _{i=1}^n(x_i-y_i{)}^2\right)}^{\frac{1}{2}}$$There is a countable basis $B\subset \mathcal{P}({\mathbb{R}}^n)$.

证明. Take $B\left(x,\frac{1}{k}\right)$, where $x\in {\mathbb{Q}}^n$, $k\in \mathbb{N}$. $\mathcal{B}$ consists of all these balls as $x$ ranges over ${\mathbb{Q}}^n$ and $k$ ranges over $\mathbb{N}$.

$U\subset {\mathbb{R}}^n$ open. Take $x\in U$. Then $\exists \epsilon >0$, s.t. $B(x,\epsilon )\subset U$. Take $y\in B\left(x,\frac{1}{3}\epsilon \right)\cap {\mathbb{Q}}^n$.

Consider $x\in B\left(y,\frac{2}{3}\epsilon \right)\subset U$. $$d(x,y)<\frac{1}{3}\epsilon$$Fix $r\in \mathbb{Q}$ with $\frac{1}{3}\epsilon <r<\frac{2}{3}\epsilon$. Then $B(y,r)\in \mathcal{B}$ and $B(y,r)\subset U$.

$\square$

1.4Topological Manifold

定义 1.4.1. A topological manifold $M$ of dimension $n\in \mathbb{N}$ is a topological space $(M,\mathcal{O})$, s.t.

(1)	$(M,\mathcal{O})$ is locally homeomorphic to ${\mathbb{R}}^n$ (“locally Euclidean”);
(2)	$(M,\mathcal{O})$ is Hausdorff;
(3)	$(M,\mathcal{O})$ has a countable basis for $\mathcal{O}$.

Let $(X,{\mathcal{O}}_X)$ and $(Y,{\mathcal{O}}_Y)$ be topological spaces.

定义 1.4.2. A map $f:X\to Y$ is continuous if $f^{-1}(U)\in {\mathcal{O}}_X$ for all $U\in {\mathcal{O}}_Y$.

定义 1.4.3. $f$ is homeomorphism if $f$ is bijective and continuous, and $f^{-1}$ is also continuous.

定义 1.4.4. $(X,{\mathcal{O}}_X)$ and $(Y,{\mathcal{O}}_Y)$ are locally homeomorphic if every $x\in X$ has an open neighborhood $U$ which is homeomorphic to an open set in $Y$.

例 1.4.5.  

(1)	$M={\mathbb{R}}^n$.
(2)	$M$ is a manifold $\Rightarrow$ any open $U\subset M$ is also a manifold.
(3)	$M$ is a manifold of dimension $m$ and $N$ is a manifold of dimension $n$ $\Rightarrow$ $M\times N$ is a manifold of dimension $m+n$.
(4)	$S^n:=\{x\in {\mathbb{R}}^{n+1}\mid ||x||=1\}$. This is a $n$-dimensional manifold.
(5)	$T^n=\underset{n\text{times}}{\underbrace{S^1\times \cdots \times S^1}}$ by (3) and (4).
(6)	Every surface is a $2$-dimensional manifold.