Topology, An Introduction

Introduction

Most people encounter a geometric definition of open and closed sets before they see the more abstract definition. Intuitively, a set is open if it approaches, but does not contain its boundary. An open ended line segment, or the points x² + y² < 1 in the plane, are open sets. A closed set includes its boundary. A double-ended segment in a line, or the points x² + y² ≤ 1 in the plane, are closed sets. If the unit disk contains half of its boundary, or a line segment contains one end point and not the other, it is neither open nor closed.

You can do an enormous amount of math without knowing any more than this. But the above isn't very rigorous, and doesn't generalize to abstract structures in algebra and geometry. So if you're ready, let's redefine open sets.

A topology is a set of points, and a collection of subsets that are designated as "open". For instance, the points inside the unit circle form a subset of the plane that is designated as open. This is one of many open sets in the plane. To be a proper topology, the empty set and the entire set must be open, and open sets remain open under finite intersection and arbitrary union. Intersect two open sets and find another open set. Take the union of lots of open sets and find another open set.

By definition, the complement of an open set is closed, and it follows that closed sets are closed under arbitrary intersection and finite union. By inference, the entire set and the empty set are closed.

A given set S, sometimes called a space, can have many topologies, depending on how you define the open sets in S. The minimum topology is S and the empty set. The maximum topology declares every subset of S open and closed.

Stronger Topology

A topology is stronger than another if it contains more open sets. The maximum topology is stronger than all other topologies. This forces a partial ordering on topologies.

Let Q be a collection of topologies on S. Define T as the intersection over all the topologies in Q. In other words, a set is open in T iff it is open in every Q_i. Verify that T satisfies the criteria for being a topology.

If R is a collection of open sets in S, there is a minimum topology T containing R. Intersect all the topologies that contain R.

Neighborhood

A neighborhood is another word for an open set. A neighborhood about the point p is an open set containing p. This term is used even in abstract settings, where a neighborhood may not be a tiny piece of real estate in the plane.