By SIDNEY A. MORRIS
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Define the mapping f : S → P(S) by f (x) = {x}, for each x ∈ S. Proof. Clearly f is a one-to-one correspondence between the sets S and f (S). So S is equipotent to the subset f (S) of P(S). 33 Proposition. If S is any infinite set, then P(S) is an uncountable set. Proof. As S is infinite, the set P(S) is infinite. 30, P(S) is not equipotent to S. Suppose P(S) is countably infinite. 10, S is countably infinite. So S and P(S) are equipotent, which is a contradiction. Hence P(S) is uncountable. 33 demonstrates the existence of uncountable sets.
Conversely, assume that for each U ∈ τ and each x ∈ U , there exists a B ∈ B with x ∈ B ⊆ U . We have to show that every open set is a union of members of B. So let V be any open set. Then for each x ∈ V , there is a Bx ∈ B such that x ∈ Bx ⊆ V . Clearly V = x∈V Bx. ) Thus V is a union of members of B. 3 Proposition. Let B be a basis for a topology τ on a set X. Then a subset U of X is open if and only if for each x ∈ U there exists a B ∈ B such that x ∈ B ⊆ U . Proof. Let U be any subset of X.