2. Show that ℝ×ℝ in the dictionary order topology is metrizable.3. Let X be a metric space with metric d.(a) Show that d: X × X →ℝ is continuous.(b) Let X' denote a space having the same underlying set as X. Show that ifd: X' × X' →ℝ is continuous, then the topology of X' is finer than thetopology of X.The Metric Topology127One can summarize the result of this exercise as follows: If X has a metric d,then the topology induced by d is the coarsest topology relative to which thefunction d is conti
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2. Show that ℝ×ℝ in the dictionary order topology is metrizable.3. Let X be a metric space with metric d.(a) Show that d: X × X →ℝ is continuous.(b) Let X' denote a space
2. Show that ℝ×ℝ in the dictionary order topology is metrizable.3. Let ...
Oct 19, 2023
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