# Connected space

A **connected space** is a topological space that cannot be split into two non-empty parts separated from each other by the topology: it admits no partition into two disjoint non-empty open sets, equivalently no non-trivial clopen subset, a set that is both open and closed at once.<sup>[1](https://webpages.ciencias.ulisboa.pt/~cabarbarosie/topology/textbook/index.shtml)</sup> The property formalizes the intuition of a space that is "all in one piece," with no isolated islands, and it is preserved under homeomorphisms.<sup>[2](https://encyclopediaofmath.org/wiki/Connectivity)</sup>

| Key fact | Detail |
|---|---|
| Definition | A space is disconnected if it has a non-trivial clopen subset, or equivalently a separation into two disjoint non-empty open sets; otherwise it is connected.<sup>[1](https://webpages.ciencias.ulisboa.pt/~cabarbarosie/topology/textbook/index.shtml)</sup> |
| Path connectedness | A space is path connected if any two points are joined by a continuous map from \( [0,1] \); path connected implies connected, but not conversely.<sup>[3](https://msl.cs.uiuc.edu/planning/node140.html)</sup> |
| Standard counterexample | The topologist's sine curve \( X = \{(x,y) \in \mathbb{R}^2 : x = 0 \text{ or } y = \sin(1/x)\} \) is connected but not path connected.<sup>[3](https://msl.cs.uiuc.edu/planning/node140.html)</sup> |
| Continuous images | The continuous image of a connected space is connected; for real-valued functions this gives the intermediate value theorem.<sup>[4](https://proofwiki.org/wiki/Continuous_Image_of_Connected_Space_is_Connected/Corollary_3)</sup> |
| Components | The component of a point is the union of all connected subsets containing it; components are closed, disjoint, and form a partition of the space.<sup>[5](https://encyclopediaofmath.org/wiki/Connected_component)</sup> |
| Local version | A space is locally connected at a point if every neighborhood of the point contains a connected neighborhood; components of a locally connected space are open-and-closed.<sup>[6](https://encyclopediaofmath.org/wiki/Locally_connected_space)</sup> |
| History | The modern definition is the Lennes–Hausdorff separation condition; Hausdorff's *Grundzüge der Mengenlehre* (1914) is the classic source through which it spread.<sup>[7](https://www.cambridge.org/core/books/abs/who-gave-you-the-epsilon/evolution-of-the-topological-concept-of-connected/FE78DBB4DAA42DB6A04EE3908DBB9D95)</sup> |

## Definition and equivalent forms

Let \( (X, \mathcal{T}) \) be a topological space. The space is **disconnected** if there is a non-trivial partition of \( X \) into open sets, that is, a subset \( A \subset X \) that is open and whose complement \( X \setminus A \) is also open. Since the complement of an open set is closed, such an \( A \) is simultaneously open and closed, a clopen set. The space is connected if it is not disconnected.<sup>[1](https://webpages.ciencias.ulisboa.pt/~cabarbarosie/topology/textbook/index.shtml)</sup> Equivalently, the only clopen subsets of a connected space are \( \emptyset \) and \( X \) itself.<sup>[8](https://www.ace335.com/328/chapters/connectedness)</sup>

A subset \( C \subset X \) is connected if it is connected in the subspace topology induced from \( X \). Connectedness of a set is an intrinsic property: if the induced topology on \( C \) is the same whether \( C \) sits in \( X \) or in another space \( Y \), then \( C \) is connected as a subset of \( X \) exactly when it is connected as a subset of \( Y \).<sup>[1](https://webpages.ciencias.ulisboa.pt/~cabarbarosie/topology/textbook/index.shtml)</sup>

Two basic closure facts follow from the definition. If \( C \subset D \subset \overline{C} \) and \( C \) is connected, then \( D \) is connected, so in particular the closure of a connected set is connected. If a family of connected sets has a non-empty intersection, then the union of the family is connected.<sup>[1](https://webpages.ciencias.ulisboa.pt/~cabarbarosie/topology/textbook/index.shtml)</sup> The second fact underlies the construction of components: the union of all connected subsets containing a given point is itself connected, and it is the largest connected subset containing that point.<sup>[5](https://encyclopediaofmath.org/wiki/Connected_component)</sup>

## Examples in Euclidean space

The Euclidean plane \( \mathbb{R}^2 \) is connected. Removing a single point leaves the remainder connected, but removing a circle that does not reduce to a point disconnects the plane into an inside and an outside.<sup>[2](https://encyclopediaofmath.org/wiki/Connectivity)</sup> The real line is connected, and \( \mathbb{R}^n \) is connected because it is the union of all straight lines through the origin, each homeomorphic to \( \mathbb{R} \), and these lines share the origin.<sup>[1](https://webpages.ciencias.ulisboa.pt/~cabarbarosie/topology/textbook/index.shtml)</sup> In a normed space, convex sets and star-shaped sets are connected.<sup>[1](https://webpages.ciencias.ulisboa.pt/~cabarbarosie/topology/textbook/index.shtml)</sup> An open connected subset is called a domain, and domains and convex subsets of Euclidean spaces are path connected.<sup>[5](https://encyclopediaofmath.org/wiki/Connected_component)</sup>

At the other extreme, every discrete topological space is totally disconnected: its only connected subsets are single points.<sup>[9](https://www.phy.olemiss.edu/~luca/Topics/top/connected.html)</sup>

## Path connectedness and the sine curve

A space is **path connected** if for any two points \( x_1, x_2 \) there is a continuous map \( \tau : [0,1] \to X \) with \( \tau(0) = x_1 \) and \( \tau(1) = x_2 \).<sup>[3](https://msl.cs.uiuc.edu/planning/node140.html)</sup> Every path connected space is connected, because the interval \( [0,1] \) is connected and its continuous image joins the two points.<sup>[3](https://msl.cs.uiuc.edu/planning/node140.html)</sup>

The converse fails. The **topologist's sine curve**

\[ X = \{(x,y) \in \mathbb{R}^2 : x = 0 \text{ or } y = \sin(1/x)\} \]

is connected but not path connected. The graph of \( \sin(1/x) \) oscillates near the \( y \)-axis with frequency tending to infinity, and after the union with the \( y \)-axis is taken, no path reaches the \( y \)-axis from the sine curve, although the space as a whole is connected.<sup>[3](https://msl.cs.uiuc.edu/planning/node140.html)</sup>

Two remedies are standard. One is to build paths directly into the definition and work with path connectedness. The other is to restrict the class of spaces considered: for manifolds, connected and path connected coincide, so the distinction disappears.<sup>[3](https://msl.cs.uiuc.edu/planning/node140.html)</sup> A stronger variant, arc connectedness, requires a path that is one-to-one, hence a homeomorphism onto its image.<sup>[5](https://encyclopediaofmath.org/wiki/Connected_component)</sup>

## Continuous images and the intermediate value theorem

Connectedness is preserved by continuous maps: if \( f : X \to Y \) is continuous and \( X \) is connected, then \( f(X) \) is connected.<sup>[4](https://proofwiki.org/wiki/Continuous_Image_of_Connected_Space_is_Connected/Corollary_3)</sup> The proof idea is to pull a hypothetical disconnection of \( f(X) \) back through \( f \), whose preimages of open sets are open, producing a disconnection of \( X \).<sup>[10](https://www2.math.upenn.edu/~gressman/analysis/04-ivt.html)</sup>

The main consequence is the intermediate value theorem. A closed real interval \( [a,b] \) is connected, so its continuous image in \( \mathbb{R} \) is connected, and the connected subsets of \( \mathbb{R} \) are exactly the intervals; a continuous real-valued function on \( [a,b] \) therefore attains every value between \( f(a) \) and \( f(b) \).<sup>[4](https://proofwiki.org/wiki/Continuous_Image_of_Connected_Space_is_Connected/Corollary_3)</sup> The same mechanism works on any connected domain in \( \mathbb{R} \), not only on a segment.<sup>[11](https://commeunjeu.org/cours/fr-mpi-compactness-connectedness-finite-dimension/?lang=en)</sup>

## Components and related notions

The **connected component** of a point \( x \) is the union of all connected subsets of \( X \) containing \( x \); it is the largest connected subset containing \( x \). Components are closed sets, distinct components are disjoint, and the family of all components partitions \( X \).<sup>[5](https://encyclopediaofmath.org/wiki/Connected_component)</sup> The **quasi-component** of a point is the intersection of all clopen subsets containing it; the component is always contained in the quasi-component, and in compact spaces the two coincide.<sup>[5](https://encyclopediaofmath.org/wiki/Connected_component)</sup>

Several strength levels sit between connectedness and discreteness. A space is totally separated (hereditarily disconnected) if all components are singletons, and totally disconnected if all quasi-components are singletons; an extremally disconnected [Hausdorff space](https://www.edgepedia.org/hausdorff-space), one in which the closure of every open set is open, is totally disconnected.<sup>[5](https://encyclopediaofmath.org/wiki/Connected_component)</sup> A connected compact space is called a **continuum**. The intersection of a decreasing family of non-empty continua is a non-empty continuum, and by Sierpiński's theorem a continuum cannot be decomposed into a countable union of non-empty disjoint closed subsets.<sup>[5](https://encyclopediaofmath.org/wiki/Connected_component)</sup>

## Local connectedness

A space is **locally connected at a point** if every neighborhood of the point contains a connected neighborhood; it is locally connected if this holds at every point.<sup>[5](https://encyclopediaofmath.org/wiki/Connected_component)</sup> In a locally connected space, every connected component is open-and-closed.<sup>[6](https://encyclopediaofmath.org/wiki/Locally_connected_space)</sup> Local connectedness and connectedness are independent: a connected space need not be locally connected, with the comb space and broom space serving as counterexamples.<sup>[9](https://www.phy.olemiss.edu/~luca/Topics/top/connected.html)</sup>

The analogous local path condition is stronger. A space is locally path connected at \( x \) if every neighborhood of \( x \) contains a path-connected neighborhood, and every locally path-connected space is locally connected, while the converse fails.<sup>[12](https://mathstrek.blog/2013/03/10/topology-locally-connected-and-locally-path-connected-spaces/)</sup> Local connectedness also does not follow from path connectedness: a wheel-like union of the circle \( S^1 \) with infinitely many spokes \( X_n = \{(t\cos(\pi/n), t\sin(\pi/n)) : n/(n+1) \le t \le 1\} \) is path connected, yet no path-connected open subset near the origin fits inside a ball of radius \( 1/2 \).<sup>[12](https://mathstrek.blog/2013/03/10/topology-locally-connected-and-locally-path-connected-spaces/)</sup>

The hypotheses interact usefully. If a space is connected and locally path connected, then it is path connected: each path component is open, its complement is a union of open path components, so every path component is clopen, and connectedness forces a single component.<sup>[12](https://mathstrek.blog/2013/03/10/topology-locally-connected-and-locally-path-connected-spaces/)</sup> A partial converse to "locally path connected implies locally connected" is the Mazurkiewicz–Moore–Menger theorem: any complete metric locally connected space is locally path connected.<sup>[6](https://encyclopediaofmath.org/wiki/Locally_connected_space)</sup> In the compact metric setting, every connected, locally connected space (a Peano continuum) is locally path connected.<sup>[13](https://math.stackexchange.com/questions/843173/path-connected-and-locally-connected-space-that-is-not-locally-path-connected)</sup>

## History

The intuitive idea of a continuum as an infinitely divisible, connected object goes back to Greek discussions of the structure of physical substances and geometric figures, and [Aristotle](https://www.edgepedia.org/aristotle) concluded that a continuum cannot be composed of points.<sup>[14](https://repozytorium.biblos.pk.edu.pl/resources/30385)</sup><sup> • </sup><sup>[15](https://www.sciencedirect.com/science/chapter/edited-volume/abs/pii/B9780444503558500623)</sup> A mathematical definition took shape in the nineteenth century in the work of Cantor and Dedekind; Cantor in 1883 defined a set \( M \) as connected when, for every positive \( \varepsilon \) and every \( a, b \in M \), finitely many points \( p_1, \dots, p_n \) can be chosen so that the successive distances \( ap_1, p_1p_2, \dots, p_nb \) are all less than \( \varepsilon \).<sup>[16](https://doi.org/10.1016/j.hm.2008.01.001)</sup>

The modern open-set definition is due to Lennes as well as Hausdorff. For years European topologists credited [Felix Hausdorff](https://www.edgepedia.org/felix-hausdorff) as the prime originator, because their knowledge of set theory and topology came from his classic *Grundzüge der Mengenlehre*, published in 1914. By the 1944 third edition Hausdorff had learned of Lennes's earlier version and called attention to it, and the definition came to be called the Lennes–Hausdorff definition; many textbooks now speak of the Hausdorff–Lennes separation condition.<sup>[7](https://www.cambridge.org/core/books/abs/who-gave-you-the-epsilon/evolution-of-the-topological-concept-of-connected/FE78DBB4DAA42DB6A04EE3908DBB9D95)</sup> The systematic study of connectedness began with that 1914 book.<sup>[17](https://www.rexresearch1.com/TopologyLibrary/PrinciplesTopologyCroom.pdf)</sup>

## References

1. [Definition 16.1, Connected spaces, Topology textbook, Universidade de Lisboa](https://webpages.ciencias.ulisboa.pt/~cabarbarosie/topology/textbook/index.shtml)
2. [Connectivity, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Connectivity)
3. [Connected vs. path connected, S. M. LaValle, Planning Algorithms](https://msl.cs.uiuc.edu/planning/node140.html)
4. [Continuous Image of Connected Space is Connected, Corollary 3, ProofWiki](https://proofwiki.org/wiki/Continuous_Image_of_Connected_Space_is_Connected/Corollary_3)
5. [Connectivity (Connected component), Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Connected_component)
6. [Locally connected space, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Locally_connected_space)
7. [R. L. Wilder, "Evolution of the Topological Concept of 'Connected'", in *Who Gave You the Epsilon?*, MAA](https://www.cambridge.org/core/books/abs/who-gave-you-the-epsilon/evolution-of-the-topological-concept-of-connected/FE78DBB4DAA42DB6A04EE3908DBB9D95)
8. [Connectedness, ACE 328/335 notes](https://www.ace335.com/328/chapters/connectedness)
9. [Topics: Connectedness in Topology, University of Mississippi](https://www.phy.olemiss.edu/~luca/Topics/top/connected.html)
10. [The Intermediate Value Theorem, Advanced Analysis, University of Pennsylvania](https://www2.math.upenn.edu/~gressman/analysis/04-ivt.html)
11. [Compactness, connectedness, finite dimension, MPI lessons](https://commeunjeu.org/cours/fr-mpi-compactness-connectedness-finite-dimension/?lang=en)
12. [Locally Connected and Locally Path-Connected Spaces, Mathematics and Such](https://mathstrek.blog/2013/03/10/topology-locally-connected-and-locally-path-connected-spaces/)
13. [Path-connected and locally connected space that is not locally path-connected, Mathematics Stack Exchange](https://math.stackexchange.com/questions/843173/path-connected-and-locally-connected-space-that-is-not-locally-path-connected)
14. [The notion of connectedness in mathematical analysis of XIX century, Politechnika Krakowska repository](https://repozytorium.biblos.pk.edu.pl/resources/30385)
15. [Connectedness, Encyclopedia of General Topology, ScienceDirect](https://www.sciencedirect.com/science/chapter/edited-volume/abs/pii/B9780444503558500623)
16. [The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica](https://doi.org/10.1016/j.hm.2008.01.001)
17. [Principles of Topology, F. H. Croom, chapter on connectedness](https://www.rexresearch1.com/TopologyLibrary/PrinciplesTopologyCroom.pdf)

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