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 topologyis not connected, or disconnected, because we can partition it asandare disjoint becausebut the set with topology is connected because we cannot writeas the union of two sets in the topology. For exampleconsisting 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 partitioninto disjoint sets, for example, but the first of these is not an open set in any topology onis not a partition because

A subset of a topological spaceis a connected set if it is a connected space when viewed as a subspace of

If we delete a point fromthen the resulting space is still connected. If we delete a line fromthen the space with the line removed is not connected. If a line is deleted fromthen 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 aplane from a spaceleaves behind a connected space.

A spacewith the discrete topology is totally disconnected. For everyin the space, there is an open setin the topology, and another setalso in the topology, with

We can extend connectness to intersections of sets.

Ifis a family of connected subsets of a topological spaceindexed by an arbitrary setsuch that for allthen is also connected. For example the set of lines,is connection sincewhere

If is a nonempty family of connected subsets of a topological spacesuch thatthen is also connected. We can again talk the set of linesas an example since