Flashcards
§5 Metric Spaces
Cards in this deck (7)
- 1Definition 5.1: MetricA \(\underline{metric}\) 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] - 2Definition: Metric SpaceA \(\underline{metric-space}\) is a pair (X,d) where d is a metric on X.
- 3Theorem 5.3In 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 - 4Definition 5.19: IsometryAn \(\underline{isometry}\) 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 \(\underline{isometric}\). - 5TheoremIf (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. - 6Definition: MetrizabilityA 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. - 7TheoremAny metrizable space in Hausdorff.
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- Diploma/Certificado de qualificação profissional de nivel 3 Mathematics (Topology (MATH313)) Flashcards on §5 Metric Spaces, created by Amy Wogan on 27/04/2016.
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