Analysis II | Part IB, 2003

Explain briefly what is meant by a metric space, and by a Cauchy sequence in a metric space.

A function d:X×XRd: X \times X \rightarrow \mathbb{R} is called a pseudometric on XX if it satisfies all the conditions for a metric except the requirement that d(x,y)=0d(x, y)=0 implies x=yx=y. If dd is a pseudometric on XX, show that the binary relation RR on XX defined by xRyd(x,y)=0x R y \Leftrightarrow d(x, y)=0 is an equivalence relation, and that the function dd induces a metric on the set X/RX / R of equivalence classes.

Now let (X,d)(X, d) be a metric space. If (xn)\left(x_{n}\right) and (yn)\left(y_{n}\right) are Cauchy sequences in XX, show that the sequence whose nnth term is d(xn,yn)d\left(x_{n}, y_{n}\right) is a Cauchy sequence of real numbers. Deduce that the function dˉ\bar{d} defined by

dˉ((xn),(yn))=limnd(xn,yn)\bar{d}\left(\left(x_{n}\right),\left(y_{n}\right)\right)=\lim _{n \rightarrow \infty} d\left(x_{n}, y_{n}\right)

is a pseudometric on the set CC of all Cauchy sequences in XX. Show also that there is an isometric embedding (that is, a distance-preserving mapping) XC/RX \rightarrow C / R, where RR is the equivalence relation on CC induced by the pseudometric dˉ\bar{d} as in the previous paragraph. Under what conditions on XX is XC/RX \rightarrow C / R bijective? Justify your answer.

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