In algebra, we defined how to operate on two elements to get another; in topology we instead define in a rough sense how close points are to each other. Specifically, beyond being a set, a topological space includes a definition of open sets or neighborhoods. This is also called defining a topology for the space.
One can instead define a numeric distance between points, forming a metric space, which is automatically a topological space; but in physics it is generally more profitable to jump from topological spaces directly to manifolds, which have a natural distance function that can always make them metric spaces.
The more modern definition of a topology uses open sets, but equivalent results can be obtained by defining neighborhoods. We will not go into foundational topics here, but it is worthwhile to be reminded of a few notions that appear frequently concerning a space \({X}\):
- Weaker (AKA coarser) topology: one topology is weaker than another if it defines less open sets, i.e., every open set in the weaker topology is also in the stronger (AKA finer) one
- Hausdorff space (AKA \({T_{2}}\), satisfying the second “separation axiom”): any two points have disjoint neighborhoods, i.e. no two points are “right next to each other”; this is a common condition on spaces to avoid pathological exceptions
- Compact space: every cover has a finite sub-cover; we will not define these terms here, but this usually translates to closed and bounded
- Connected space: the only subsets that are both open and closed are \({X}\) and the empty set; this implies the intuitive idea that \({X}\) does not consist of several disjoint pieces
- Continuous mapping \({f\colon X\to Y}\): inverse images of open sets are open, i.e. if \({B\subset Y}\) is open, \({f^{-1}\left(B\right)\subset X}\) is open; this implies the intuitive properties of continuity
- Path-connected space: a more restrictive kind of connectivity that for any two points requires the existence of a continuous mapping from \({\mathbb{R}}\) that passes through them, i.e. there is a path connecting any two points
- Homeomorphism: a continuous bijective mapping with a continuous inverse; the most basic equivalency in topology
A topological n–manifold (AKA \({n}\)-manifold) is then defined to be a Hausdorff space in which every point has an open neighborhood homeomorphic to an open subset of \({\mathbb{R}^{n}}\). A topological n–manifold with boundary allows the neighborhoods of points to be homeomorphic to an open subset of the closed half of \({\mathbb{R}^{n}}\), i.e. of the portion of \({\mathbb{R}^{n}}\) on one side of and including \({\mathbb{R}^{n-1}}\). The points that map to \({\mathbb{R}^{n-1}}\) form the boundary of the manifold, and are a \({\left(n-1\right)}\)-manifold. A compact manifold without boundary is often called a closed manifold to distinguish it from a compact manifold with boundary. Most definitions of a manifold usually also include some additional technical requirement to avoid pathological exceptions (e.g. second countable \({\Rightarrow}\) paracompact \({\Leftrightarrow}\) metrizable, none of which we will define here).
Topological spaces include many interesting objects that do not lie on our path towards manifolds, for example objects constructed by taking the limit of some iterative process. In this book we will be mainly interested in results that are valid for spaces with the “nicer” manifold structure. For example, for manifolds the definitions of path-connected and connected are equivalent.