Point-set Topology

a.k.a. General Topology

A.1 Topological Spaces

Map $f:\mathbb{R}^n\to\mathbb{R}^m$ is continuous iff inverse image $f^-1(V)$ of open set $V\in\mathbb{R}^m$ is open in $\mathbb{R}^n$.

$d(p,q)=[\sum_{i=1}^n (p^i-q^i)^2]^{\frac{1}{2}}$ where points $p,q\in\mathbb{R}$.

$B(p,r)={x\in\mathbb{R}^n\|d(x,p)<r}$ with center $r\in\mathbb{R}^n$ and radius $r>0$.

a topology is a collection $\mathcal{T}$ of subset of set $S$ (the open sets) such that

$\emptyset,S\in\mathcal{T}$
Arbitrary unions of open sets are open: $U_\alpha\in\mathcal{T}, \alpha\in\mathcal{J}\Rightarrow \bigcup_{\alpha\in\mathcal{J}} U_\alpha$ for index set $\mathcal{J}$
Finite intersections of open sets are open: $U_1, …,U_n \in \mathcal{T}\Rightarrow U_1\cap,…,\cap U_n \in \mathcal{F}$

Definitions

open sets. The elements of a topology.
topological space. The pair $(S,\mathcal{T})$ for set $S$ and topology $\mathcal{T}$.
neighborhood of a point p on S. an open set $U$ containing $p$.
courser topology. for $\mathcal{T}_1 : \mathcal{T}_1 \subset \mathcal{T}_2$.
finer topology. for $\mathcal{T}_2 : \mathcal{T}_1 \subset \mathcal{T}_2$.
standard topology. collection of open sets $U \in \mathbb{R}^n$ such that $\forall p\in U, \exists \text{ open ball } B(p,\varepsilon) \subset U$.

For topological space S, subset $A\in S$ is open $\Leftrightarrow \forall p \in A, \exists$ open set $V : p \in V \subset A$.

definitions

trivial or indiscrete topology. the coarsest topology consisting of only ${\emptyset,S}$.
discrete topology. a topology $\mathcal{T} : \mathcal{T}$ contains all subsets of $S$.
discrete space. a topological space with a discrete topology.
singleton set. a set with a single element.
closed set. the complement of an open set.
Finite-completement topology. Topology composed of closed sets containing $\emptyset,S$ with finite unions and arbitrary intersections.

Subspace $\mathcal{T}_A := {A\cap U: U \in \mathcal{T}}$ for topology $\mathcal{T}$ and $A\subset S$, which is topology itself.

defintions.

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