## Connected Spaces

A connected space is a topological space that cannot be represented as the union of disjoint nonempty open subsets. If a space can be so represented, it is disconnected. Connectedness is one of the principal topological properties used to distinguish topological spaces.

For example the space with the topology is not connected, or disconnected, because we can partition it as  and are disjoint because but the set with topology is connected because we cannot write as the union of two sets in the topology. For example consisting of the set of matrices with non - zero determinant is not connected because we can partition it into the two sets, one set consisting of matrices with positive determinant and the other consisting of matrices with negative determinant. Every space with the discrete topology is disconnected, while every space with the indiscrete topology is connected.

Examples: is connected. We can partition into disjoint sets, for example, but the first of these is not an open set in any topology on  is not a partition because A subset of a topological space is a connected set if it is a connected space when viewed as a subspace of If we delete a point from then the resulting space is still connected. If we delete a line from then the space with the line removed is not connected. If a line is deleted from then the space with the line removed is connected. In higher dimensions we can find a route around the deleted line using the other dimension. In general, removing a plane from a space leaves behind a connected space.

A space with the discrete topology is totally disconnected. For every in the space, there is an open set in the topology, and another set also in the topology, with We can extend connectness to intersections of sets.

If is a family of connected subsets of a topological space indexed by an arbitrary set such that for all then is also connected. For example the set of lines, is connection since where If is a nonempty family of connected subsets of a topological space such that then is also connected. We can again talk the set of lines as an example since  