Semi-Riemannian Geometry

Chapter 9 Topology

9.1 Topology

Having completed an overview of linear and multilinker algebra, we now turn our attention to topology. This is an extensive area of mathematics, and of necessity the coverage presented here is highly selective. To the uninitiated, topology can be dauntingly abstract. The best example of a topology (and the motivation for much of what follows) is the Euclidean topology on ℝ ^m , covered briefly in Section 9.4 and preceded by preparatory material in Section 9.2 and Section 9.3. Readers new to topology might find it helpful to peruse these sections early on to get a glimpse of where the discussion below is heading.

A topology on a set X is a collection 𝒯 of subsets of X such that:

• [T1] ∅ and X are in 𝒯.
• [T2] The union of any sub collection of elements of 𝒯 is in 𝒯.
• [T3] The intersection of any finite sub collection of elements of 𝒯 is in 𝒯.

The pair (X, 𝒯) is referred as a topological space and each element of 𝒯 is said to be an open set in 𝒯 or simply open in 𝒯. Each element of X is called a point in X. Any open set in 𝒯 containing a given point x in X is said to be a neighborhood of x in X . We say that a subset K of X is a closed set in 𝒯 or simply closed in 𝒯 if is an open set in X. It is often convenient to adopt the shorthand of referring to X as a topological space, with 𝒯 understood from the context. Accordingly, if U is an open ...