Connected space

In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. 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.

Path connectedness

The space X is said to be path-connected if for any two points x and y in X there exists a continuous function f from the unit interval to X with f(0) = x and f(1) = y.

Related Topics:
Continuous function - Unit interval

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(This function is called a path from x to y.)

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Every path-connected space is connected.

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Example of connected spaces that are not path-connected include the extended long line L* and the topologist's sine curve.

Related Topics:
Long line - Topologist's sine curve

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However, subsets of the real line R are connected if and only if they are path-connected; these subsets are the intervals of R.

Related Topics:
Real line - If and only if - Intervals

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Also, open subsets of Rn or Cn are connected if and only if they are path-connected.

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Additionally, connectedness and path-connectedness are the same for finite topological spaces.

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A space X is said to be arc-connected if any two distinct points can be joined by an arc, that is a path f which is a homeomorphism between the unit interval and its image f(). It can be shown any Hausdorff space which is path-connected is also arc-connected. An example of a space which is path-connected but not arc-connected is provided by adding a second copy 0' of 0 to the nonnegative real numbers [0,∞). One endows this set with a partial order by specifying that 0'<a for any positive number a, but leaving 0 and 0' incomparable. One then endows this set with the order topology, that is one takes the open intervals

Related Topics:
Homeomorphism - Hausdorff space - Partial order

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(a,b)={x | a<x<b} and the half-open intervals [0,a)={x | 0≤x<a}, [0',a)={x | 0'≤x<a} as a base for the topology. The resulting space is a T1 space but not a Hausdorff space. Clearly 0 and 0' can be connected by a path but not by an arc in this space.

Related Topics:
Base - T₁ - Hausdorff space

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