# Topology, Lecture 1

Flashcards by Jörg Schwartz, updated more than 1 year ago
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Flashcards on Topology, Lecture 1, created by Jörg Schwartz on 01/17/2016.

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 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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