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| Question | Answer |
| Definition 5.1: Metric | A \(\underline{metric}\) on a set X is a function d:X×X⟶\(ℝ_{≥0}\)={x ∈ℝ: x≥0} s.t. - d(x,y)=0 iff x=y - d(x,y)=d(y,x) [symmetry] - d(x,z) ≤ d(x,y) + d(y,z) [triangle inequality] |
| Definition: Metric Space | A \(\underline{metric-space}\) is a pair (X,d) where d is a metric on X. |
| Theorem 5.3 | In any (X,d) the set \(B_{d}\)={\(B_{d}\)(x,ε) : x∈X & ε>0} of all open balls is a \(\underline{basis}\) for the topology \(T_d\) on X |
| Definition 5.19: Isometry | An \(\underline{isometry}\) f:(X,d)⟶(y,d') is a bijection from X onto Y s.t. ∀x,x'∈X d(x,x')=d'(\(f_x\),\(f_{x'}\)) [distance preserving] If such f exists then (X,d) & (Y,d') are \(\underline{isometric}\). |
| Theorem | If (X.d) is isometric to (X',d') then (X,\(T_d\)) is homeomorphic to (X',\(T_{d'}\)) under the isometry. So isometry is a stronger notion than homeomorphism. |
| Definition: Metrizability | A top. space (X,T) is \(\underline{metrizable}\) if there exists a metric d on X s.t. T= the topology \(T_d\) induced by d. Note: Any subspace of a metrizable space itself is metrizable. |
| Theorem | Any metrizable space in Hausdorff. |
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