Topology, Lecture 1

Question	Answer
Define a $\textit{base}$ for a topology on a set $X$	A $\textit{base}$ is a collection $\mathcal{B}\subseteq X$ , such that: $\forall x\in X\; \exists B\in \mathcal{B}\colon x\in B,$ $\forall B_1, B_2\in \mathcal{B} \text{ and } x\in B_1\cap B_2\colon \exists B_3\in B_1\cap B_2 \text{ with } x\in B_3$
Given a sequence of points in a topological space $X$ , define $\textit{convergence to the point } x\in X$	$(x_n)_{n\geq 1}$ converges to $x\in X$ if $\forall U\in \mathcal{T}\text{ with } x\in U \;\exists N\;\forall N\geq n\colon x_n\in U$
Given two topologies $\mathcal{T}_1,\mathcal{T}_2$ , define $\textit{coarser/finer}$	If $\mathcal{T}_1\subseteq\mathcal{T}_2$ , then $\mathcal{T}_1 \text{ is coarser then }\mathcal{T}_2,$ $\mathcal{T}_2 \text{ is finer then }\mathcal{T}_1$
Give a definition of a topology on a set $X$ in terms of open sets. Define $\textit{topological space}$	A topology on a set $X$ is a collection $\mathcal{T}$ of open subsets of $X$ , such that finite intersection of open sets and infinte unions of open sets are again open. The pair $(X, \mathcal{T})$ is called a $\textit{topological space}$
Define a $\textit{metric}$ (distance function) on a set $X$	A metric is a function $d\colon X\times X\rightarrow \Re_{\geq 0}$ such that for all $x,y,z\in X$ : $d(x,y) = 0\Leftrightarrow x = y$ $d(x,y) = d(y,x)$ $d(x,y) + d(y,z)\geq d(x,z)$

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	Created by Jörg Schwartz over 9 years ago