Connected Space

Connected Space

Connected and disconnected subspaces of R²

The green space A at top is simply connected whereas the blue space B below is not connected. The pink space C at top and the orange space D are both connected; C is also simply connected while D is not.

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, which is a space where any two points can be joined by a path.

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

As an example of a space that is not connected, one can delete an infinite line from the plane. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with a closed annulus removed, as well as the union of two disjoint open disks in two-dimensional Euclidean space.

Read more about Connected Space:  Formal Definition, Examples, Path Connectedness, Arc Connectedness, Local Connectedness, Theorems, Graphs, Stronger Forms of Connectedness

Famous quotes containing the words connected and/or space:

    “Nothing fortuitous happens in a child’s world. There are no accidents. Everything is connected with everything else and everything can be explained by everything else.... For a young child everything that happens is a necessity.”
    —John Berger (b. 1926)

    “But alas! I never could keep a promise. I do not blame myself for this weakness, because the fault must lie in my physical organization. It is likely that such a very liberal amount of space was given to the organ which enables me to make promises, that the organ which should enable me to keep them was crowded out. But I grieve not. I like no half-way things. I had rather have one faculty nobly developed than two faculties of mere ordinary capacity.”
    —Mark Twain [Samuel Langhorne Clemens] (1835–1910)