K-10 - Manifold

Abstract

An n-manifold is a Hausdorff space in which each point has a neighborhood homeomorphic to ℝn Euclidean n-space, and similarly an n-manifold with boundary is a Hausdorff space in which each point has a closed neighborhood homeomorphic to the unit n-ball Bn defined as {x ∈ ℝn: ║x║ ≤ 1}. Open subsets of n-manifolds (with boundary) are n-manifolds (with boundary); the Cartesian product of any n-manifold (with boundary) and any m-manifold (with boundary) is an (n +m)- manifold (with boundary). Being locally compact, all n-manifolds Mn are regular spaces. The long ray ñ—the Cartesian product of the ordinal space [0, ω1) with [0, 1) under the lexicographical ordering ñ—is a normal but non-metrizable 1-manifold with boundary. There exists a separable 2-manifold W2 that contains an uncountable, discrete, closed subset, hence W2 cannot even be normal. Continuum Hypothesis (CH) implies the existence of a nonmetrizable, perfectly normal manifold. This chapter shows that the combination of Martin's Axiom and the negation of CH imply the metrizability of any perfectly normal manifold. It is proved that CH implies the existence of a perfectly normal 4-manifold W4 whose covering dimension is larger than 4.