Pseudometric spaces with topologies

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1. Let (X, d) and (Y, e) be pseudometric spaces with topologies Td and Te metrized by d and e respectively. Let f be a function from X into Y . Show that the following are equivalent (as stated in the first paragraph of this chapter):

(a) f is continuous: f -1(U ) ∈ Td for all U ∈ Te .

(b) f is sequentially continuous: for every x ∈ X and every sequence xn → x for d, we have f (xn ) → f (x ) for e.

2. Let (S, d) be a metric space and X a subset of S. Let the restriction of d to X × X also be called d. Show that the topology on X metrized by d is the same as the relative topology of the topology metrized by d on S.

Reference no: EM13990638

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