## Approach to Defining a Topology

Here, I will be discussing my perspective on the motivation for Topology. The following information comes from my notes on Topology found here.

We are motivated by the following definitions and theorems:

• Definition. (Open and Closed) A set $G\subset\mathbb{R}$ is open if $\forall{x}\in{G} \ \exists\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\subset{G}$ (interior points). A set $F\subset\mathbb{R}$ is closed if $F^c=\mathbb{R}-F$ is open.
• Ex) (a) Let $[0,1]\subset\mathbb{R}$. Then to be an open set, $\forall{x}\in{G} \ \exists\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\subset[0,1]$. However, if we chose $0$ or $1$ for $x$, there is no $(x-\epsilon,x+\epsilon)$ in $[0,1]$. Hence, $[0,1]$ is not an open set. On the other hand, we define $[0,1]^c=\mathbb{R}-[0,1]$, which is open. Therefore, by the definition of a closed set, $[0,1]$ is a closed set. (b) Let $(0,1)\subset\mathbb{R}$. Then we take $\forall{x}\in(0,1)$, $\exists\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\subset(0,1)$. We can see that all points $x$ contained in $(0,1)$ have a $(x-\epsilon,x+\epsilon)$-interval. I.e., for $x=0.0001, \ (0.00001,0.001)\subset(0,1)$. Therefore, $(0,1)$ is an open set.
• Theorem. (Open Sets) (a) If $\{G_{\lambda}:\lambda\in\Lambda\}$ is a collection of open sets, then $\cup_{\lambda\in\Lambda}$ is open. (b) If $\{G_k:1\leq{k}\leq{n}\}$ is a finite collection of open sets, then $\cap^n_{k=1}G_k$ is open. (c) Both $\emptyset$ and $\mathbb{R}$ are open.
• Theorem. (Closed Sets) (a) If $\{F_{\lambda}:\lambda\in\Lambda\}$ is a collection of closed sets, then $\cap_{\lambda\in\Lambda}F_{\lambda}$ is closed(b) If $\{F_k:1\leq{k}\leq{n}\}$ is a finite collection of closed sets, then $\cup_{k=1}^n{F_k}$ is closed(c) Both $\emptyset$ and $\mathbb{R}$ are closed.
• Definition. (Limit Points) $x_0$ is a limit point of $S\subset\mathbb{R}$ if $\forall\epsilon>0$, the $(x_0-\epsilon,x_0+\epsilon)\cap{S}\backslash\{x_0\}\neq\emptyset$.

From this, we can form our definition of a topology.