Connectivity

connectedness

The property of a topological space stating that it is impossible to represent the space as the sum of two parts separated from each other, or, more precisely, as the sum of two non-empty disjoint open-closed subsets. A space that is not connected is called disconnected. For example, the usual Euclidean plane is a connected space; if a point is removed, then the remainder is connected, but if a circle that does not reduce to a point is removed, then the remainder is disconnected.

The abstract property of connectivity expresses the intuitive notion of a connected space as an entity in which there are no isolated "islands" . The connectivity of a topological space is preserved under homeomorphisms, and is one of the most important properties of topological spaces.

A subset of a topological space is called connected if it is connected in the subspace topology. Once this concept is introduced one can assert that a space is connected if any two points of it lie in some connected subset, that is, if they can be joined by some connected set. From this point of view the abstract property of connectivity can be regarded as a generalization of path connectivity, that is, the property of a space whereby any two of its points can be joined by a path (a continuous image of a segment). An open connected subset is called a domain. Domains and convex subsets in Euclidean spaces are path connected, and hence, connected.

If a family of connected subsets has a non-empty intersection, then the union of the sets of this family is a connected set. For every point of a topological space the union of all connected subsets containing the point is the largest connected subset containing it; this union is called the component of this point. Components are closed sets, and different components are disjoint.

The quasi-component of a point is the intersection of all open-closed subsets that contain this point. The component of a point is contained in the quasi-component of this point. In compact spaces, components and quasi-components coincide.

A space is called totally separated (dispersed; hereditarily disconnected) if all its components are singletons, that is, if all connected subsets consist of one point. A space is called totally disconnected (nowhere connected) if all its quasi-components are singletons. A space is called extremally disconnected if the closure of any open set is open. An extremally-disconnected Hausdorff space is totally disconnected, and every totally-disconnected space is hereditarily disconnected. There is a connected space that contains a dispersion point the removal of which leaves a totally-disconnected space; an example is the Kuratowski–Knaster fan.

A connected compact space is called a continuum. The intersection of a decreasing family of non-empty continua is a non-empty continuum. But a continuum cannot be decomposed into the union of a countable family of non-empty disjoint closed subsets (Sierpiński's theorem).

A space is said to be irreducible between two of its points if it is connected and if the two points cannot be joined by a connected set other than the whole space. Every continuum contains, for any two points, a subcontinuum that is irreducible between them (the Mazurkiewicz–Janicewski theorem).

A space is called locally connected at a point if any neighbourhood of this point contains a connected neighbourhood of this point; a locally connected space is locally connected at every point.

A space is called connected in dimension $n$ if every continuous mapping of the $n$-dimensional sphere into it can be extended to a continuous mapping of the $(n+1)$-dimensional ball. The property of being connected in dimension 1 is equivalent to the fundamental group of the space being trivial.

A continuous mapping of one topological space into another is called monotone if the inverse image of each point is a connected subset. For closed maps the property of being monotone is equivalent to the pre-image of each connected subset being connected.

References

[1]	P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2]	K. Kuratowski, "Topology" , 2 , Acad. Press (1968) (Translated from French)

Comments

Path connectedness is also called linear connectedness or arcwise connectedness. An irreducible space is one that cannot be written as a union of two proper closed subsets. The algebraic geometric spectrum $\mathrm{Spec}(A)$ of a ring $A$ is irreducible if and only if the nil radical of $A$ is prime.

References

[a1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)

Comments

A separation in a topological space is a non-trivial open-closed subset. A space is thus disconnected if and only if it has a separation, and totally separated if for any points $x \neq y$ there is a separation containing $x$ and not $y$.

The equivalence relation on a space $X$ that $x \equiv y$ if and only if for every separation $S$ of $X$ either $x$ and $y$ are in $S$ or $x$ and $y$ are in $X \setminus S$ defines equivalence classes called quasicomponents. The quasicomponent of $x$ is the intersection of all open-closed sets of $X$ containing $x$. If there is only one quasicomponent in $X$ then it is connected. A space is totally separated if and only if its quasicomponents are all singletons.

The equivalence relation on a space $X$ that $x \equiv y$ if and only if there is a connected subspace of $X$ containing $x$ and $y$ defines as classes the connected components: these are the maximal connected subspaces of $X$. The component of $x$ is the union of all connected subsets of $X$ containing $x$. A space is totally disconnected if and only if its connected components are all singletons.

A topological space is irreducible or hyperconnected if it is not the union of two proper closed subsets, or equivalently no two non-empty open subsets are disjoint, or again any non-empty open set is dense. It is ultraconnected if no two non-empty closed subsets are disjoint or equivalently the closures of singletons are never disjoint. These two properties are independent, although each implies connectedness, and an ultraconnected space is path-connected.

A topological space is absolutely connected if it cannot be expressed as a non-trivial union of disjoint closed subsets.

A space $X$ is path-connected if for any two points $x,y \in X$ there is a continuous map $p$ from the unit interval $[0,1]$ with the usual topology to $X$ such that $p(0) = x$ and $p(1) = y$. A space is arc-connected if the path can be taken to be one-to-one.

References

[b1]	Steen, Lynn Arthur; Seebach, J.Arthur jun. Counterexamples in topology (2nd ed.) Springer (1978) ISBN 0-387-90312-7 Zbl 0386.54001