Q1. Suppose (X, d) is a metric space. Define a function by ρ : ...

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Q1. Suppose (X, d) is a metric space. Define a function by ρ : X × X → R, ρ(x, y) := min(d(x, y), 1).
(a) Show that the topology generated by ρ equals the topology generated by d.
Q2. Suppose (X, d) is a metric space with the metric topology.
(a) Show that d : X × X → R is continuous.
(b) Show that if T’ is any other topology on X in which d is continuous, then the metric topology is coarser than T’.

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