Topology, Lecture 1

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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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