Topology

Definition

$\mathcal{T}\subset 2^X$

A topology $\mathcal{T}$ is a collection of subsets of a set $X$ (subset of the power set of the set) satisfying the following conditions:

• $\varnothing ,X\in \mathcal{T}$
• $\forall \{U_i{\}}_{i=1}^{\infty }\subset \mathcal{T},\bigcup _{i=1}^{\infty }{U}_i\in \mathcal{T}$
• $\forall \{U_i{\}}_{i=1}^n\subset \mathcal{T},\bigcap _{i=1}^n{U}_i\in \mathcal{T}$

An element of a topology $U\in \mathcal{T}$ is called an Open Set.

Examples

• Trivial (smallest) topology: $\mathcal{T}=\{\varnothing ,X\}$
• Discrete (largest) topology: $\mathcal{T}=2^X$
• Standard Topology on $\mathbb{R}$

Examples on Real Number

• Trivial topology: ${\mathcal{T}}_t=\{\varnothing ,\mathbb{R}\}$
• Discrete topology: ${\mathcal{T}}_d=2^{\mathbb{R}}$
• Standard topology: ${\mathcal{T}}_s$ generated by ${\mathcal{B}}_s=\{(a,b)\subset \mathbb{R}|a<b\}$
• Lower limit topology: ${\mathcal{T}}_l$ generated by ${\mathcal{B}}_l=\{[a,b)\subset \mathbb{R}|a<b\}$
• K-topology: ${\mathcal{T}}_K$ generated by ${\mathcal{B}}_K=\{(a,b),(a,b)-K\subset \mathbb{R}|a<b\}$ where $K=\{\frac{1}{n}|n\in \mathbb{N}\}$
• Finite complement topology: ${\mathcal{T}}_f=\{U\subset \mathbb{R}|U=\varnothing \text{or}|\mathbb{R}-U|<\infty \}$

flowchart TD
 
Tt["$T_t$"]
Td["$T_d$"]
Ts["$T_s$"]
Tl["$T_l$"]
TK["$T_K$"]
Tf["$T_f$"]
 
Td --- Tl
Td --- TK
Tl --- Ts
TK --- Ts
Ts --- Tf
Tf --- Tt