Let X be any set with discrete metric (d(x;y) = 1 if x 6= y and d(x;y) = 0 if x= y), and let Y be an arbitrary metric space. $$c))(a)" Analogous to the proof of \(a))(c)". Prove that fx ngconverges if and only if it is eventually constant, that is, there … The so-called taxicab metric on the Euclidean plane declares the distance from a point (x, y) to a point (z, w) to… 5. Here, the distance between any two distinct points is always 1. A metric space (X,d) is a set X with a metric d deﬁned on X. The only non-trivial bit is the triangle inequality, but this is also obvious. If = then it Proof: Let U {\displaystyle U} be a set. We can deﬁne many diﬀerent metrics on the same set, but if the metric on X is clear from the context, we refer to X as a metric space and omit explicit mention of the metric d. Example 7.2. is a metric. However, we can also deﬁne metrics in all sorts of weird and wonderful ways Example 1 The discrete metric. Then it is straightforward to check (do it!) This, in particular, shows that for any set, there is always a metric space associated to it. This sort of proof is hard to explain without knowing exactly what your particular definitions are. Solution: (M1) to (M4) can be checked easily using de nition of the discrete metric. Example 5. The discrete metric, where (,) = if = and (,) = otherwise, is a simple but important example, and can be applied to all sets. This distance is called a discrete metric and (X;d) is called a discrete metric space. Let be any non-empty set and deﬁne ( ) as ( )=0if = =1otherwise then form a metric space. we need to show, that if x ∈ U {\displaystyle x\in U} then x {\displaystyle x} is an internal point. For every space with the discrete metric, every set is open. (a) Let fx ngbe a sequence in X. 9. Let me present Jered Wasburn-Moses’s answer in a slightly different way. Because this is the discrete metric \(\displaystyle \left( {\forall t \in X} \right)\left[ {B_{1/2} \left( t \right) = \{ t\} } \right]$$. First, recall that a function f: X!R from a set Xto R is bounded if there is some M2R such Proof. However, here is some general guidance. Show that the discrete metric is in fact a metric. Proof. Show that Xconsists of eight elements and a metric don Xis de ned by d(x;y) = Page 4 After the standard metric spaces Rn, this example will perhaps be the most important. Other articles where Discrete metric is discussed: metric space: …any set of points, the discrete metric specifies that the distance from a point to itself equal 0 while the distance between any two distinct points equal 1. that dis a metric on X, called the discrete metric. (Hamming distance) Let X be the set of all ordered triples of zeros and ones. 10. 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