• Topology
  • open
  • topological space $(M,\mathcal{O})$
  • neighborhood
  • closure
    • interior point
    • exterior point
    • boundary point
  • limit point
    • sequence convergence
  • continuity
  • Brouwer’s fixed point theorem

Topology

set $M$

set $\mathcal{O}\subseteq \mathcal{P}(M)$ with all its element open

• point, $p\in M$

open

1. $\varnothing \in \mathcal{O},M\in \mathcal{O}$
2. $\forall U_i,U_j\in \mathcal{O},U_i\cap U_j\in \mathcal{O}$
3. $\forall U_i,U_j\in \mathcal{O},U_i\cup U_j\in \mathcal{O}$

• closed, $V\subseteq M$ where $V^C\in \mathcal{O}$
  • $\varnothing ,M$ are both open and closed

topological space $(M,\mathcal{O})$

set $M$ and choice of set of open subset $\mathcal{O}$

• chaotic topology, ${\mathcal{O}}_c=\{\varnothing ,M\}$
• discrete topology, ${\mathcal{O}}_d=P(M)$

neighborhood

neighborhood of $p$: $U\in \mathcal{O},p\in U$

• deleted neighborhood, $\hat{U}=U\setminus \{p\}$

closure

$S\subseteq M$

closure of $S$, $\bar{S}=S\cup \partial S$

interior point

point $p$

$\exists U\ni p$ s.t. $U\cap S=U$

• interior of $S$, set of all interior point of $S$

exterior point

point $p$

$\exists U\ni p$ s.t. $U\cap S=\varnothing$

• exterior of $S$, set of all exterior point of $S$

boundary point

neither interior nor exterior of $S$

• $\partial S$, boundary of $S$, set of all boundary point of $S$

limit point

limit point $p$ of $S\iff$

$\forall \hat{U},\hat{U}\cap S\ne \varnothing$

sequence convergence

$\forall U\ni p,\exists m\in \mathbb{N}$ s.t. $n>m\Rightarrow s(n)\in U$

• in chaotic topology, every sequence converge to every point

continuity

topological space $(M,{\mathcal{O}}_M),(N,{\mathcal{O}}_N)$

map $f:M\to N$ continuous $\iff$

$$\forall V\in {\mathcal{O}}_M,{\text{preim}}_f(V)\in {\mathcal{O}}_M$$

Brouwer’s fixed point theorem

continuous $g:D\to D$

$\Rightarrow \exists$ at least one fixed point

assign each fixed point index ${\nu }_i$

$\Rightarrow$ Lefschetz number $L={\sum }_i{\nu }_i$ is topological invariant