Every topological space can be decomposed into disjoint maximal connected subspaces, called its connected components. The underlying set of a topological space is the disjoint union of the underlying sets of its connected components, but the space itself is not necessarily the coproduct of its connected components in the category of spaces.
One often studies topological ideas first for connected spaces and then generalises to general spaces. This is especially true if one is studying such nice topological spaces that every space is a coproduct of connected components (such as for example locally connected spaces; see below).
preserves coproducts. It's actually enough to require that it preserves binary coproducts (a detailed proof in a more general setting is given at connected object); in that case, notice that we always have a map
Here are some equivalent ways to say that is connected in more elementary terms: * Whenever , where the right side is the coproduct of spaces (so that are identified with disjoint open subspaces of ), then exactly one of is inhabited (so the other is empty, making the inhabited one homeomorphic to ). * If is clopen (both closed and open), then if and only if is inhabited.
Many authors allow the empty space to be connected. You can get this concept from the elementary definitions above by changing ‘exactly one’ to ‘at most one’ and changing ‘if and only if’ to ‘if’. Categorially, this version of connectedness requires only that the maps
be surjections. However, many results come out more cleanly by disqualifying the empty space (much as one disqualifies when one defines the notion of prime number). See also the discussion at empty space and too simple to be simple.
The elementary definitions above have been carefully phrased to be correct in constructive mathematics. One may also see classically equivalent forms that are constructively weaker.
Wide pushouts of connected spaces are connected. (This would of course be false if the empty space were considered to be connected.) This follows from the hom-functor definition of connectedness, plus the fact that coproducts in commute with wide pullbacks. More memorably: connected colimits of connected spaces are connected.
The interval , as a subspace of , is connected. (This is the topological underpinning of the intermediate value theorem.)
If is a connected subspace and (i.e. if is between and its closure), then is connected.
The basic results above give a plethora of ways to construct connected spaces. More exotic examples are sometimes useful, especially for constructing counterexamples.
where is some chosen fixed irrational number. It is easy to see this space is Hausdorff (using the fact that is irrational). However, the closure of consists of points of with either or , in other words, the union of two infinitely long strips of width and slopes , . Clearly any two such closures intersect, and therefore the space is connected.
This example is due to Golomb. Topologize the set of natural numbers by taking a basis to consist of sets , where are relatively prime. The space is Hausdorff, but the intersection of the closures of two non-empty open sets is never empty, so this space is connected.
Every topological space admits an equivalence relation where means that and belong to some subspace which is connected. The equivalence class of an element is thus the union of all connected subspaces containing ; it follows readily from Result 2 that is itself connected. It is called the connected component of . It is closed, by Result 5. A space is connected if and only if it has exactly one connected component (or at most one, if you allow the empty space to be connected).
There is another equivalence relation where if for every continuous mapping to a discrete space . The equivalence class of may be alternatively described as the intersection of all clopens that contain . This is called the quasi-component of , denoted here as . It is easy to prove that
and that equality holds if is compact Hausdorff or is locally connected (see below), but also in other circumstances (such as the space of rational numbers as a topological subspace of the real line).
For an example where , take to be the following subspace of :
In this example, , but .
An entry point is given in the following remark/warning:
It is not generally true that a space is the coproduct (in ) of its connected components, nor of its quasi-components. For example, the connected components in Cantor space (with its topology as a product of 2-point discrete spaces) are just the singletons, but the coproduct of the singleton subspaces carries the discrete topology; another example with this feature is the set of rational numbers with its absolute-value topology (the one induced as a topological subspace of the real line).
A space is locally connected if every open set, as a topological space, is the coproduct (in ) of its connected components. Equivalently, a space is locally connected if every point has a neighborhood basis of connected open sets.
In a locally connected space, every connected component is clopen; in particular, connected components and quasi-components coincide. We warn that connected spaces need not be locally connected; for example, the topologist’s sine curve of Example 4 is connected but not locally connected.
Examples of locally connected spaces include topological manifolds.
Let be the full inclusion of locally connected spaces. The following result is straightforward but useful.
is a coreflective subcategory of , i.e., the inclusion has a right adjoint . For a given space, has the same underlying set as , topologized by letting connected components of open subspaces of generate a topology.
Being a coreflective category of a complete and cocomplete category, the category is also complete and cocomplete. Of course, limits and particularly infinite products in are not calculated as they are in ; rather one takes the limit in and then retopologizes it according to Theorem 1. (For finite products of locally connected spaces, we can just take the product in – the result will be again locally connected.)
Let be the underlying set functor, and let be the functors which assign to a set the same set equipped with the codiscrete and discrete topologies, respectively. Let be the functor which assigns to a locally connected space the set of its connected components.
There is a string of adjoints
and moreover, the functor preserves finite products.
The proof is largely straightforward; we point out that the continuity of the unit is immediate from a locally connected space’s being the coproduct of its connected components. As for preserving finite products, write locally connected spaces , as coproducts of connected spaces
then their product in coincides with their product in , and is
where each summand is connected by Result 3. From this it is immediate that preserves finite products.
A space is totally disconnected if its connected components are precisely the singletons of .
In other words, a space is totally disconnected if its coreflection into is discrete. Such spaces recur in the study of Stone spaces.
For future reference, we record the following result.
A quotient space of a locally connected space is also locally connected.
Suppose is a quotient map, and let be an open neighborhood of . Let be the connected component of in ; we must show is open in . For that it suffices that be open in , or that each is an interior point. Since is locally connected, the connected component of in is open, and the subset is connected, and therefore (as is the maximal connected subset of containing ). Hence , proving that is interior to , as desired.
The conclusion does not follow if is merely surjective. For example, there is a surjective (continuous) map from to the Warsaw circle, but the latter is not locally connected.
An important variation on the theme of connectedness is path-connectedness. If is a space, define the path component to be the subspace of all for which there exists a continuous map where , .
Observe that this is a reflexive coequalizer, as witnessed by the mutual right inverse .
(We can even topologize by taking the coequalizer in of
We say is path-connected if it has exactly one path component.
It follows easily from the basic results above that each path component is connected. However, it need not be closed (and therefore need not be the connected component of ); see the following example. The path components and connected components do coincide if is locally path-connected.
The topologist’s sine curve
provides a classic example where the path component of a point need not be closed. (Specifically, consider a point on the locus of .)
Finally, as a contrast to a path-connected space, a totally path-disconnected space is a space such that its set of path components is equal to the underlying set of the space. Equivalently, that there are no non-constant paths. This by far does not mean that the space is discrete!
A refinement of the notion of path-connected space is that of arc-connected (or arcwise-connected) space:
A space is arc-connected if for any two distinct there exists an injective continuous map such that and .
Arc-connected spaces are of course path-connected, but there are trivial examples (using an indiscrete topology) that the converse fails to hold. A rather nontrivial theorem is the following:
A path-connected Hausdorff space is arc-connected.
This immediately generalizes to the statement that in a Hausdorff space , any two points that can be connected by a path can be connected by an arc: just apply the theorem to the image .
For a proof of this theorem, see Willard, theorem 31.2. More precisely, that result states that a Peano space, i.e., a compact, connected, locally connected, and metrizable space, is arc-connected if it is path-connected. It then suffices to observe that the continuous image of a path is in fact a Peano space, so that the path can be replaced by an arc.
If is Hausdorff and there is a continuous surjection , then is a Peano space.
Obviously is compact (Hausdorff) and connected. is a quotient space of , since is a closed surjection (using compactness of and Hausdorffness of ), and therefore is locally connected by Lemma 1. Being compact Hausdorff, is regular, so to show metrizability it suffices by the Urysohn metrization theorem to show is second-countable.
Let be a countable base for and let be the collection consisting of finite unions of elements of . We claim is an (evidently countable) base for . Indeed, suppose is open and ; then is compact, so there exist finitely many with
Put . The first inequality is equivalent to by the adjunction . The second inequality implies , where the equality , equivalent to , follows from surjectivity of . Thus we have shown is a base.
The converse of this lemma is the celebrated Hahn-Mazurkiewicz theorem:
Let be a nonempty Hausdorff space. Then there exists a continuous surjection if is a Peano space. In particular, a nonempty Peano space is path-connected.
(The terminology “Peano space” is given in recognition of Peano’s discovery of space-filling curves, as for example the unit square.)
As above, let be the functor which assigns to each space its set of path components .
The functor preserves arbitrary products.
Let be a family of spaces; we must show that the comparison map
is invertible. Injectivity: suppose are tuples that map to the same tuple of path-components ; we must show that and belong to the same path component. For each , both and belong to , so we may choose a path connecting to . Then connects to . (Note this uses the axiom of choice.) Surjectivity: for any tuple , the component is nonempty for each , so we may choose an element therein. Then maps to . Again this uses the axiom of choice.
An elegant proof of the previous proposition but for preservation of finite products is as follows: both and preserve products, and a reflexive coequalizer of product-preserving functors , being a sifted colimit, is also product-preserving.
The functor preserves arbitrary coproducts.
The functor preserves coproducts since is connected, and similarly for . The coequalizer of a pair of natural transformations between coproduct-preserving functors is also a coproduct-preserving functor.
Point-set topology is filled with counterexamples. An unusual type of example is that of pseudo-arc:
A pseudo-arc is a metric continuum with more than one point such that every subcontinuum (a subspace that is a continuum) cannot be expressed as a union of two proper subcontinua.
A pseudo-arc is necessarily totally path-disconnected: two distinct points of cannot be connected by a path in . Indeed, the image of such a path would be a path-connected Hausdorff space, hence arc-connected by Theorem 3. Letting be an arc from to , we have that the continuum is a union of proper subcontinua and , a contradiction. Thus, a pseudo-arc is an example of a compact connected metrizable space that is totally path-disconnected.
Remarkably, all pseudo-arcs are homeomorphic, and a pseudo-arc is a homogeneous space. Perhaps also remarkable is the fact that the collection of pseudo-arcs in the Hilbert cube (or in any Euclidean space) is a dense set (see G-delta set) in the Polish space of all nonempty compact subsets of under the Hausdorff metric; see Bing2, theorem 2.
There is a function such that the limit of the diagram
is a pseudo-arc.
Roughly speaking, Henderson’s is a small “notched” perturbation of the squaring function , as illustrated on page 38 (of 58) here.
A space in which the only connected subspaces are the singletons and the empty set is called totally disconnected space.
Examples of countable connected Hausdorff spaces were give in
Material on arc-connected spaces and the Hahn-Mazurkiewicz theorem can be found in Chapter 31 of
Material on pseudo-arcs can be found in