By SIDNEY A. MORRIS
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Clearly the restriction of g −1 to ST is a bijection of ST onto TT . Finally, the restriction of f to S∞ is a bijection of S∞ onto T∞ . Define h : S → T by f (s) h(s) = g −1(s) f (s) if s ∈ SS if s ∈ ST if s ∈ S∞ . Then h is a bijection of S onto T . So S is equipotent to T . 35 Our next task is to define what we mean by “cardinal number”. 2 Definitions. A collection, ℵ, of sets is said to be a cardinal number if it satisfies the conditions: (i) Let S and T be sets. If S and T are in ℵ, then S ∼ T ; (ii) Let A and B be sets.
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. Assume that for each x ∈ U , there exists a Bx ∈ B such that x ∈ Bx ⊆ U . Clearly U = x∈U Bx. 3 is precisely what we used in defining the Euclidean topology on R. We said that a subset U of R is open if and only if for each x ∈ U , there exist a and b in R with a < b, such that x ∈ (a, b) ⊆ U.