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implies d 1 d 2 . And if d 1 , d 2 are obtained from norms • i , i = 1, 2 then by the preceding exercise we have d 1 d 2 ⇔ • 1 • 2 ⇔ (2.8). But if at least one of the metrics d 1 , d 2 does not come from a norm, equivalence d 1 d 2 does not imply (2.8): Consider X = R with d 1 (x, y) = |x -y| and d 2 (x, y) = max(1, d 1 (x, y)). Then d 1 d 2 by Exercise 2.2.14, but (2.8) cannot hold since d 1 is unbounded and d 2 is bounded. 2 Definition 2.2.18 The topology on R n (and C n ) defined by the equivalent norms • p , p ∈ [1, ∞] is called the usual or Euclidean topology. We see that passing from a metric space (X, d) to the topological space (X, τ d ), we may lose information. This actually is one of the main reasons for working with topological spaces, since even when all spaces in sight are metrizable, the actual choice of the metrics may be irrelevant and therefore distracting! Purely topological proofs tend to be cleaner than metric proofs. ### 2.3 Some standard topologies It is time to see some topologies that do not come from a metric! Some standard topologies can actually be defined on any set X: Definition/Proposition 2.3.1 Let X be a set. Then the following are topologies on X:

Who reads Topology for the Working Mathematician?

Author

Muger M.

Published

2022

Language

EN

Category

nonfiction

Subjects

Mathematics, Geometry And Topology, Stem