Publisher Summary This chapter discusses the theory of nonmetrizable manifolds. Manifolds have been considered too specialized, and the distinctions between them are too fine and subtle for the techniques of point-set topology to give interesting results about them. These manifolds share many basic properties with the spaces they mimic, making it possible to bring the tools of set theory into play. The structure theory of manifolds provides a fertile ground for the application of set-theoretic concepts and techniques. Manifolds also provide a convenient central category in the study of topological spaces. Manifolds can be locally compact, locally separable, locally connected, locally Lindelof, locally countable chain condition, first countable, locally second countable, locally metrizable, and completely regular. A manifold is a connected space for which there is a positive integer n such that each point has a neighborhood that is homeomorphic to Rn. A manifold is a Moore space if and only if it is submetacompact. Every submetacompact manifold is subparacompact. A manifold is metrizable if and only if it is paracompact.

The Theory of Nonmetrizable Manifolds

A strong regularity condition is given which generalizes the notion of a completely regular mapping defined by Dyer and Hamstrom. These mappings are shown to be fiber spaces of various types with some additional hypotheses. Strongly regular mappings on manifolds whose point inverses are ANR or AR are considered. For example, it is shown that with AR fibers the map is either a homeomorphism or the manifold has boundary which meets all non- degenerate point inverses.

A strong regularity condition on mappings

Let $X$ be an $(n+1)$-dimensional manifold, $\Delta$ be a one-dimensional foliation on $X$, and $p: X \to X / \Delta$ be a quotient map. We will say that a leaf $\omega$ of $\Delta$ is special whenever the space of leaves $X / \Delta$ is not Hausdorff at $\omega$. We present necessary and sufficient conditions for the map $p: X \to X / \Delta$ to be a locally trivial fibration under assumptions that all leaves of $\Delta$ are non-compact and the family of all special leaves of $\Delta$ is locally finite.

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One-dimensional foliations on topological manifolds

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Louis F. McAuley

Papers

Completely regular mappings, fiber spaces, the weak bundle properties, and the generalized slicing structure properties