1. For all j in J where J is some indexing set, let U, be a...

 

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1. For all j in J where J is some indexing set, let U, be a topological space, let V; be a subspace of Uj. Define U = IjUj with the product topology. JEJ Give V the subspace topology where V but finitely many je J} = {(xj)j€J € U\xj € V; for all a. Assume that V₁ is compact and U, is locally compact for every j ¤ J. Prove that V must be locally compact. b. Assume that V; is connected and U, is locally connected for every je J. Prove that V must be locally connected. c. Assume that U; is second-countable for every j E J and is also T3. Prove that V must be metrizable. 1. For all j in J where J is some indexing set, let U, be a topological space, let V; be a subspace of Uj. Define U = IjUj with the product topology. JEJ Give V the subspace topology where V but finitely many je J} = {(xj)j€J € U\xj € V; for all a. Assume that V₁ is compact and U, is locally compact for every j ¤ J. Prove that V must be locally compact. b. Assume that V; is connected and U, is locally connected for every je J. Prove that V must be locally connected. c. Assume that U; is second-countable for every j E J and is also T3. Prove that V must be metrizable.

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a Assume that V is compact and U is locally compact for every j J Prove that V must be locally compact Proof Since U is locally compact for every j J ... View the full answer