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1. True. Note that in the subspace topology {0,1}, both {0} and {1} are open. Thus, the product topologies X×{0} and X×{1} are both open and disjoint. Since X×{0,1}=(X×{0})∪(X×{1}), then X×{0,1} is disconnected.
2. False. One counter example is the space R with discrete topology. It is induced by the discrete metric but is not finite....

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