Poincare Project: Connectedness, Closed Sets and Topological Properties

Some topological spaces have the property that they can be decomposed into two disjoint non-empty open sets. In other words, there exist two non-empty open sets whose intersection is empty but whose union is the entire space. Take our ball of clay and cut it in half.

Such a topological space is said to be disconnected. Topological spaces for which this is not true are said to be connected.

Another way of defining the same notion of connectedness is via the notion of closed sets. (The existence of open sets suggested there would be something called closed sets right?)

A closed set of a topological space is simple one whose complement is open. In other words, if you have an open set, then the set of points not in that open set is a closed set. One interesting property of this definition is it allows a set to be both open and closed at the same time. If a set and its complement are both open, then both sets are also closed.

Because, by definition, the empty set and the set of all points in a topological space are open sets, they are also closed sets. And here is where we come to the definition of connectedness based on closed sets.

A topological space is connected if and only if the only two sets that are both open and closed are the empty set and the set of all points. If any other sets are both open and closed then the topological space must be disconnected.

It is fairly easy to see why this is true. If two disjoint non-empty open sets A and B have a union which is the entire space then A and B are each others complements. Therefore A must be closed (because B is open) and B must be closed (because A is open). Therefore A and B are both open and closed.

Connectedness is said to be a topological property because it is based purely on the open sets and no additional structure. Because topological properties are based only on the open sets, they are preserved by a homeomorphism. All homeomorphisms preserve all topological properties. So if a space is connected, then any space homeomorphic to it will also be connected. An important corollary is that you can never find a homeomorphism between a connected space and a disconnected one, or between any two spaces that have differing topological properties.

In the example of cutting our ball of clay in half, the before and after are not homeomorphic because the before is connected and the after is disconnected. Again, we've ripped apart points that were once in lots of open sets together so that now the only open set they share is the topological space as a whole.

UPDATE: next post