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 are used to distinguish topological spaces.

- Formal definition
- Connected components
- Disconnected spaces
- Examples
- Path connectedness
- Arc connectedness
- Local connectedness
- Set operations
- Theorems
- Graphs
- Stronger forms of connectedness
- See also
- References
- Further reading

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

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.

A topological space *X* is said to be **disconnected** if it is the union of two disjoint non-empty open sets. Otherwise, *X* 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.

For a topological space *X* the following conditions are equivalent:

*X*is connected, that is, it cannot be divided into two disjoint non-empty open sets.*X*cannot be divided into two disjoint non-empty closed sets.- The only subsets of
*X*which are both open and closed (clopen sets) are*X*and the empty set. - The only subsets of
*X*with empty boundary are*X*and the empty set. *X*cannot be written as the union of two non-empty separated sets (sets for which each is disjoint from the other's closure).- All continuous functions from
*X*to are constant, where is the two-point space endowed with the discrete topology.

Historically this modern formulation of the notion of connectedness (in terms of no partition of *X* into two separated sets) first appeared (independently) with N.J. Lennes, Frigyes Riesz, and Felix Hausdorff at the beginning of the 20th century. See ^{ [1] } for details.

The maximal connected subsets (ordered by inclusion) of a non-empty topological space are called the **connected components** of the space. The components of any topological space *X* form a partition of *X*: they are disjoint, non-empty, and their union is the whole space. 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), which are not open.

Let be the connected component of *x* in a topological space *X*, and be the intersection of all clopen sets containing *x* (called quasi-component of *x*.) Then where the equality holds if *X* is compact Hausdorff or locally connected.

A space in which all components are one-point sets is called totally disconnected. Related to this property, a space *X* is called **totally separated** if, for any two distinct elements *x* and *y* of *X*, there exist disjoint open sets *U* containing *x* and *V* containing *y* such that *X* is the union of *U* and *V*. Clearly, any totally separated space is totally disconnected, but the converse does not hold. For example take two copies of the rational numbers **Q**, 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.

- The closed interval in the standard subspace topology is connected; although it can, for example, be written as the union of and the second set is not open in the chosen topology of
- The union of and is disconnected; both of these intervals are open in the standard topological space
- is disconnected.
- A convex subset of
**R**^{n}is connected; it is actually simply connected. - A Euclidean plane excluding the origin, 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.
- ℝ, The space of real numbers with the usual topology, is connected.
- If even a single point is removed from ℝ, the remainder is disconnected. However, if even a countable infinity of points are removed from , where the remainder is connected. If
*n*≥ 3, then remains simply connected after removal of countably many points. - Any topological vector space, e.g. any Hilbert space or Banach space, over a connected field (such as or ), 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.
^{ [2] } - 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.
- If a space
*X*is homotopy equivalent to a connected space, then*X*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 (that is, the group of
*n*-by-*n*real, invertible matrices) consists of two connected components: the one with matrices of positive determinant and the other of negative determinant. In particular, it is not connected. In contrast, 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
^{ [3] }- The spectrum of a commutative ring
*R*is connected - Every finitely generated projective module over
*R*has constant rank. *R*has no idempotent (i.e.,*R*is not a product of two rings in a nontrivial way).

- The spectrum of a commutative ring

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.

A **path-connected space** is a stronger notion of connectedness, requiring the structure of a path. A ** path ** from a point *x* to a point *y* in a topological space *X* is a continuous function *ƒ* from the unit interval [0,1] to *X* with *ƒ*(0) = *x* and *ƒ*(1) = *y*. A **path-component** of *X* is an equivalence class of *X* under the equivalence relation which makes *x* equivalent to *y* if there is a path from *x* to *y*. The space *X* is said to be **path-connected** (or **pathwise connected** or **0-connected**) if there is exactly one path-component, i.e. if there is a path joining any two points in *X*. Again, many authors exclude the empty space (note however that by this definition, the empty space is not path-connected because it has zero path-components; there is a unique equivalence relation on the empty set which has zero equivalence classes).

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 *L** and the * topologist's sine curve *.

Subsets of the real line **R** are connected if and only if they are path-connected; these subsets are the intervals of **R**. Also, open subsets of **R**^{n} or **C**^{n} are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for finite topological spaces.

A space *X* is said to be ** arc-connected** or

Every Hausdorff space that 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 of to the nonnegative real numbers One endows this set with a partial order by specifying that for any positive number but leaving and incomparable. One then endows this set with the order topology. That is, one takes the open intervals and the half-open intervals as a base for the topology. The resulting space is a T_{1} space but not a Hausdorff space. The points and can be connected by a path but not by an arc in this space.

A topological space is said to be ** locally connected at a point***x* if every neighbourhood of *x* contains a connected open neighbourhood. It is **locally connected** if it has a base of connected sets. It can be shown that a space *X* is locally connected if and only if every component of every open set of *X* is open.

Similarly, a topological space is said to be **locally path-connected** if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about **R**^{n} and **C**^{n}, 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 intervals in , such as .

A classical example of a connected space that is not locally connected is the so called topologist's sine curve, defined as , with the Euclidean topology induced by inclusion in .

The **intersection** of connected sets is not necessarily connected.

The **union** of connected sets is not necessarily connected, as can be seen by considering .

Each ellipse is a connected set, but the union is not connected, since it can be partitioned to two disjoint open sets and .

This means that, if the union is disconnected, then the collection can be partitioned to two sub-collections, such that the unions of the sub-collections are disjoint and open in (see picture). This implies that in several cases, a union of connected sets *is* necessarily connected. In particular:

- If the common intersection of all sets is not empty (), then obviously they cannot be partitioned to collections with disjoint unions. Hence
*the union of connected sets with non-empty intersection is connected*. - If the intersection of each pair of sets is not empty () then again they cannot be partitioned to collections with disjoint unions, so their union must be connected.
- If the sets can be ordered as a "linked chain", i.e. indexed by integer indices and , then again their union must be connected.
- If the sets are pairwise-disjoint and the quotient space is connected, then X must be connected. Otherwise, if is a separation of X then is a separation of the quotient space (since are disjoint and open in the quotient space).
^{ [4] }

The set difference of connected sets is not necessarily connected. However, if and their difference is disconnected (and thus can be written as a union of two open sets and ), then the union of with each such component is connected (i.e. is connected for all ).

Proof ^{ [5] } |
---|

By contradiction, suppose is not connected. So it can be written as the union of two disjoint open sets, e.g. . Because is connected, it must be entirely contained in one of these components, say , and thus is contained in . Now we know that: The two sets in the last union are disjoint and open in , so there is a separation of , contradicting the fact that is connected. |

**Main theorem of connectedness**: Let*X*and*Y*be topological spaces and let*ƒ*:*X*→*Y*be a continuous function. If*X*is (path-)connected then the image*ƒ*(*X*) is (path-)connected. This result can be considered a generalization of the intermediate value theorem.- Every path-connected space is 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 have path connected subsets, namely those subsets for which every pair of points has a path of edges joining them. But it is not always possible to find a topology on the set of points which induces the same connected sets. The 5-cycle graph (and any *n*-cycle with *n* > 3 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 ( Muscat & Buhagiar 2006 ). 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.

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

- If there exist no two disjoint non-empty open sets in a topological space,
*X*,*X*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. Note however, that 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, note that 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.

In topology and related areas of mathematics, a **product space** is the Cartesian product of a family of topological spaces equipped with a natural topology called the **product topology**. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, **open sets** are a generalization of open intervals in the real line. In a metric space—that is, when a distance is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In topology and related branches of mathematics, **separated sets** are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces as well as to the separation axioms for topological spaces.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In mathematics, an **order topology** is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.

In topology, a **clopen set** in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of *open* and *closed* are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open *and* closed, and therefore clopen.

In topology and related areas of mathematics, the **quotient space** of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the **quotient topology**, that is, with the finest topology that makes continuous the canonical projection map. In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.

In mathematics, more specifically topology, a **local homeomorphism** is a function between topological spaces that, intuitively, preserves local structure. If is a local homeomorphism, is said to be an **étale space** over Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.

In topology and related branches of mathematics, a **totally disconnected space** is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets. In every topological space, the singletons are connected; in a totally disconnected space, these are the *only* connected proper subsets.

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In general topology and related areas of mathematics, the **disjoint union** of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the **disjoint union topology**. Roughly speaking, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other.

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

In mathematics, the **particular point topology** is a topology where a set is open if it contains a particular point of the topological space. Formally, let *X* be any set and *p* ∈ *X*. The collection

In topology, a branch of mathematics, the **ends** of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the **end compactification**.

In topology and other branches of mathematics, a topological space *X* is **locally connected** if every point admits a neighbourhood basis consisting entirely of open, connected sets.

In the mathematical field of topology, a **hyperconnected space** or **irreducible space** is a topological space *X* that cannot be written as the union of two proper closed sets. The name *irreducible space* is preferred in algebraic geometry.

In topology and related areas of mathematics, a subset *A* of a topological space *X* is called **dense** if every point *x* in *X* either belongs to *A* or is a limit point of *A*; that is, the closure of *A* constitutes the whole set *X*. Informally, for every point in *X*, the point is either in *A* or arbitrarily "close" to a member of *A* — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it.

In mathematics, a **scattered space** is a topological space *X* that contains no nonempty dense-in-itself subset. Equivalently, every nonempty subset *A* of *X* contains a point isolated in *A*.

- ↑ Wilder, R.L. (1978). "Evolution of the Topological Concept of "Connected"".
*American Mathematical Monthly*.**85**(9): 720–726. doi:10.2307/2321676. - ↑ George F. Simmons (1968).
*Introduction to Topology and Modern Analysis*. McGraw Hill Book Company. p. 144. ISBN 0-89874-551-9. - ↑ Charles Weibel, The K-book: An introduction to algebraic K-theory
- ↑ Brandsma, Henno (February 13, 2013). "How to prove this result involving the quotient maps and connectedness?".
*Stack Exchange*. - ↑ Marek (February 13, 2013). "How to prove this result about connectedness?".
*Stack Exchange*.

- Munkres, James R. (2000).
*Topology, Second Edition*. Prentice Hall. ISBN 0-13-181629-2. - Weisstein, Eric W. "Connected Set".
*MathWorld*. - V. I. Malykhin (2001) [1994], "Connected space",
*Encyclopedia of Mathematics*, EMS Press - Muscat, J; Buhagiar, D (2006). "Connective Spaces" (PDF).
*Mem. Fac. Sci. Eng. Shimane Univ., Series B: Math. Sc*.**39**: 1–13. Archived from the original (PDF) on 2016-03-04. Retrieved 2010-05-17..

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