In each case, determine if the statement is true or false. If it is...

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In each case, determine if the statement is true or false. If it is true, provide a proof. If it is false, provide a counterexample.
Unless otherwise specified, spaces have standard topologies ( ℝ and ℝ 2 have the standard topologies, metric spaces have the metric topology, finite spaces have the discrete topology, product spaces have the product topology, etc.). You may assume in all cases that X is a nonempty set which contains more than one point.

1) For any nonempty space X, X × {0,1} is disconnected.
2) If the metric topology on X is discrete, then X is a finite set.
3) X is disconnected if there exists a continuous surjection f: X → {0,1}.
4) PPℝ0 is a compact space.
5) A subset A of X is called nowhere dense if Int (Cl (A)) = ∅ . If A is closed in X and X A − is dense in X, then A is nowhere dense.
6) Let ( X d, ) be a metric space with A X ⊆ and p X A ∈ − . If d p A ({ }, 0 ) = , then p A ∈ ∂ .
7) Every connected component of a compact space is compact.
8) The collection B = ⊆ {A X A| is finite} is a basis for a topology on X.
9) If X is Hausdorff and A is a compact subset of X, then ∂A is compact in X.
10) If a sequence in PPXp is convergent, then the sequence is a constant sequence.

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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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