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Wikipedia / Connected space

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 non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces.

is a if it is a connected space when viewed as a subspace.

Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is locally connected, which neither implies nor follows from connectedness.

Definition

A topological space is said to be is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.

into two separated sets) first appeared (independently) with N.J. Lennes, Frigyes Riesz, and Felix Hausdorff at the beginning of the 20th century. See for details.

Connectedness defines an equivalence relation is an equivalence relation.

Connected components

will once again be a connected subset. The connected component of a point The maximal connected subsets (ordered by inclusion) of a non-empty topological space are called the connected components form a partition of: they are disjoint, non-empty and their union is the whole space. In fact, a connected component is the same as an equivalence class when two points are equivalent if they belong to the same connected subset (see ).

Every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the rational numbers are the one-point sets (singletons. Thus each component is a one-point set.

be the intersection of all clopen (called quasi-component is compact Hausdorff or locally connected.

Disconnected spaces

A space in which all components are one-point sets is called totally disconnected is called , there exist disjoint open sets, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.

Examples

in the standard subspace topology
is disconnected.
.
.
A convex subset is connected; it is actually simply connected.
A Euclidean plane is connected, but is not simply connected. The three-dimensional Euclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensional Euclidean space without the origin is not connected.
A Euclidean plane with a straight line removed is not connected since it consists of two half-planes.
, the space of real numbers with the usual topology, is connected.
The Sorgenfrey line is disconnected.
remains simply connected after removal of countably many points.
Any topological vector space, e.g. any Hilbert space or Banach space), is simply connected.
Every discrete topological space with at least two elements is disconnected, in fact such a space is totally disconnected. The simplest example is the discrete two-point space.
On the other hand, a finite set might be connected. For example, the spectrum of a discrete valuation ring consists of two points and is connected. It is an example of a Sierpiński space.
The Cantor set is totally disconnected; since the set contains uncountably many points, it has uncountably many components.
is homotopy equivalent is itself connected.
The topologist's sine curve is an example of a set that is connected but is neither path connected nor locally connected.
The general linear group is connected. More generally, the set of invertible bounded operators on a complex Hilbert space is connected.

The spectra of commutative local ring and integral domains are connected. More generally, the following are equivalent

is connected
Every finitely generated projective module has constant rank.
has no idempotent is not a product of two rings in a nontrivial way).

An example of a space that is not connected is a plane with an infinite line deleted from it. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with an annulus removed, as well as the union of two disjoint closed disks, where all examples of this paragraph bear the subspace topology induced by two-dimensional Euclidean space. == Path connectedness ==

A is a stronger notion of connectedness, requiring the structure of a path. A path in a topological space from the unit interval. A is an equivalence class under the equivalence relation. is said to be path-connected (or pathwise connected-connected) if there is exactly one path-component.

(to see the equivalence, note a path-component in the previous sense is path-connected).

Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended long line and the topologist's sine curve.

Subsets of the real line are connected if and only if they are path-connected; these subsets are the intervals are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for finite topological spaces.== Arc connectedness == is said to be arc-connected or arcwise connected if any two topologically distinguishable points can be joined by an arc, which is an embedding. An arc-component; or equivalently an equivalence class of the equivalence relation of whether two points can be joined by an arc or by a path whose points are topologically indistinguishable.

Every Hausdorff space that is path-connected is also arc-connected; more generally this is true for a \Delta-Hausdorff space, which is a space where each image of a path is closed. An example of a space which is path-connected but not arc-connected is given by the line with two origins can be connected by a path but not by an arc.

be the line with two origins. The following are facts whose analogues hold for path-connected spaces, but do not hold for arc-connected spaces:

Continuous image of arc-connected space may not be arc-connected: for example, a quotient map from an arc-connected space to its quotient with countably many (at least 2) topologically distinguishable points cannot be arc-connected due to too small cardinality.
has two overlapping arc-components.
is not.
has two arc-components.
intersect, but their union is not arc-connected.

== Local connectedness ==

A topological space is said to be locally connected contains a connected open neighbourhood. It is locally connected if it has a base is open.

Similarly, a topological space is said to be , each of which is locally path-connected. More generally, any topological manifold is locally path-connected. Locally connected does not imply connected, nor does locally path-connected imply path connected. A simple example of a locally connected (and locally path-connected) space that is not connected (or path-connected) is the union of two separated.

A classic example of a connected space that is not locally connected is the so-called topologist's sine curve, with the Euclidean topology induced.

Set operations

The intersection of connected sets is not necessarily connected.

The union.

(see picture). This implies that in several cases, a union of connected sets necessarily connected. In particular:

), then obviously they cannot be partitioned to collections with disjoint unions. Hence the union of connected sets with non-empty intersection is connected.
) then again they cannot be partitioned to collections with disjoint unions, so their union must be connected.
, then again their union must be connected.
If the sets are pairwise-disjoint and the quotient space is connected, then is a separation of are disjoint and open in the quotient space).

The set difference).

Theorems

Main theorem of connectedness is (path-)connected. This result can be considered a generalization of the intermediate value theorem.
Every path-connected space is connected.
In a locally path-connected space, every open connected set is path-connected.
Every locally path-connected space is locally connected.
A locally path-connected space is path-connected if and only if it is connected.
The closure of a connected subset is connected. Furthermore, any subset between a connected subset and its closure is connected.
The connected components are always closed (but in general not open)
The connected components of a locally connected space are also open.
The connected components of a space are disjoint unions of the path-connected components (which in general are neither open nor closed).
Every quotient of a connected (resp. locally connected, path-connected, locally path-connected) space is connected (resp. locally connected, path-connected, locally path-connected).
Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).
Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locally path-connected).
Every manifold is locally path-connected.
Arc-wise connected space is path connected, but path-wise connected space may not be arc-wise connected
Continuous image of arc-wise connected set is arc-wise connected.

Graphs

Graphss have path connected subsets, namely those subsets for which every pair of points has a path of edges joining them. However, it is not always possible to find a topology on the set of points which induces the same connected sets. The 5-cycle odd) is one such example.

As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected sets to connected sets . Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are precisely the finite graphs.

However, every graph can be canonically made into a topological space, by treating vertices as points and edges as copies of the unit interval (see topological graph theory#Graphs as topological spaces). Then one can show that the graph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.

Stronger forms of connectedness

There are stronger forms of connectedness for topological spaces, for instance:

must be connected, and thus hyperconnected spaces are also connected.
Since a simply connected space is, by definition, also required to be path connected, any simply connected space is also connected. If the "path connectedness" requirement is dropped from the definition of simple connectivity, a simply connected space does not need to be connected.
Yet stronger versions of connectivity include the notion of a contractible space. Every contractible space is path connected and thus also connected.

In general, any path connected space must be connected but there exist connected spaces that are not path connected. The deleted comb space furnishes such an example, as does the above-mentioned topologist's sine curve.

Weaker forms

A well-chained space is connected,).