Properties of the real line and weak forms of the Axiom of Choice
... and f ((0, a)) = a. (If f ((0, a)) = a, take x0 = f ((1, a)). Then x0 = a since f is one-to-one.) For i = v + 1, let xi = f ((1, xv )). Since f is one-to-one, it is straightforward to verify that xi = xj for all i, j ∈ ω, i = j. (ii) In Solovay’s model M5(ℵ), Form 169 is false. Thus, AC(R) fails ...
... and f ((0, a)) = a. (If f ((0, a)) = a, take x0 = f ((1, a)). Then x0 = a since f is one-to-one.) For i = v + 1, let xi = f ((1, xv )). Since f is one-to-one, it is straightforward to verify that xi = xj for all i, j ∈ ω, i = j. (ii) In Solovay’s model M5(ℵ), Form 169 is false. Thus, AC(R) fails ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.