Connected space

Definition

Equivalent definitions in tabular format

A topological space is said to be connected if it satisfies the following equivalent conditions.

No.	Shorthand	A topological space is termed connected if ...	A topological space $X$ is termed connected if ...
1	Absence of two open subset separation	it cannot be expressed as a disjoint union of two nonempty open subsets.	it is not possible to find open subsets $U,V\subseteq X$ such that $U,V$ are both nonempty, $U\cap V$ is empty, and $U\cup V=X$.
2	Absence of two closed subset separation	it cannot be expressed as a disjoint union of two nonempty closed subsets.	it is not possible to find closed subsets $A,B\subseteq X$ such that $A,B$ are both nonempty, $A\cap B$ is empty, and $A\cup B=X$.
3	Absence of open subset separation	it cannot be expressed as a union of a collection of pairwise disjoint nonempty open subsets that has size more than one.	it is not possible to write $X$ as a union $\bigcup _{i\in I}U_{i}$ where $U_{i}$ are all nonempty and open, $U_{i}\cap U_{j}$ is empty for $i,j\in I,i\neq j$, and $I$ has size greater than one.
4	Absence of finite closed subset separation	it cannot be expressed as a union of a collection of finitely many pairwise disjoint nonempty closed subsets that has size more than one.	it is not possible to write $X$ as a union $\bigcup _{1\leq i\leq n}A_{i}$ where $A_{i}$ are all nonempty and closed, $A_{i}\cap A_{j}$ is empty for $1\leq i,j\leq n,i\neq j$, and $n\geq 1$.
5	Absence of proper nonempty clopen subset	the only clopen subsets of the space are the whole space and the empty subset.	for any subset $A\subseteq X$ such that $A$ is clopen, i.e., $A$ is both open and closed, either $A$ is empty or $A=X$.

The term is typically used for non-empty topological spaces. Whether the empty space can be considered connected is a moot point.

Examples

Basic examples

• The one-point space is a connected space.
• Euclidean space is connected. More generally, any path-connected space, i.e., a space where you can draw a line from one point to another, is connected. In particular, connected manifolds are connected.
• In algebraic geometry, the Zariski topology is connected.

Non-examples

• Any discrete space of size more than one is not connected.
• Totally disconnected spaces, like the set of rational numbers, is not connected, despite points being "close" to one another.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subspace-hereditary property of topological spaces	No	connectedness is not hereditary	It is possible to have a connected space $X$ and a subset $A$ of $X$ such that $A$ is not connected in the subspace topology.
weakly hereditary property of topological spaces	No	connectedness is not weakly hereditary	It is possible to have a connected space $X$ and a closed subset $A$ of $X$ such that $A$ is not connected in the subspace topology.
product-closed property of topological spaces	Yes	connectedness is product-closed	Suppose $X_{i},i\in I$, are all connected spaces. Then, the Cartesian product $\prod _{i\in I}X_{i}$ is also a connected space with the product topology.
box product-closed property of topological spaces	No	connectedness is not box product-closed	It is possible to have $X_{i},i\in I$ all connected spaces such that the Cartesian product $\prod _{i\in I}X_{i}$ is not connected in the box topology.
coarsening-preserved property of topological spaces	Yes	connectedness is coarsening-preserved	If $X$ is connected under a topology $\tau$, it remains connected when we pass to a coarser topology than $\tau$.
continuous image-closed property of topological spaces	Yes	connectedness is continuous image-closed	If $X$ is a connected space and $Y$ is the image of $X$ under a continuous map, then $Y$ is also connected.
connected union-closed property of topological spaces	Yes	connectedness is connected union-closed
closure-preserved property of topological spaces	Yes	connectedness is closure-preserved	Suppose $A$ is a subset of $X$ that is connected in the subspace topology. Then, the closure ${\overline {A}}$ is also connected in its subspace topology.

Facts

Any topological space (not necessarily connected) can be partitioned into its connected components. The space is connected iff it has a single connected component, namely the whole space itself.

This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spaces

View other homotopy-invariant properties of topological spaces OR view all properties of topological spaces

This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

This article is about a basic definition in topology.
VIEW: Definitions built on this | Facts about this | Survey articles about this
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Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
path-connected space	path joining any two points	path-connected implies connected	connected not implies path-connected	|FULL LIST, MORE INFO
simply connected space	path-connected, trivial fundamental group			|FULL LIST, MORE INFO
contractible space	homotopy-equivalent to a point			|FULL LIST, MORE INFO
irreducible space				|FULL LIST, MORE INFO
ultraconnected space				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
space with finitely many connected components				|FULL LIST, MORE INFO
space with finitely many quasicomponents				Space with finitely many connected components|FULL LIST, MORE INFO
space in which all connected components are open				Space with finitely many connected components|FULL LIST, MORE INFO
space in which the connected components coincide with the quasicomponents				Space in which all connected components are open|FULL LIST, MORE INFO

Relation with size of space

Combining connectedness with a separation axiom usually yields a lower bound on the cardinality of the space as long as it has at least two points. Below are some examples of such facts:

Other property	What its combination with being connected gives us if it has at least two points	Proof
T1 space	infinite space. In fact, any finite T1 space must be discrete	connected and T1 with at least two points implies infinite
regular Hausdorff space	uncountable space	connected and regular with at least two points implies uncountable
Urysohn space	uncountable space, cardinality at least that of the continuum	connected and Urysohn with at least two points implies cardinality at least that of the continuum
normal Hausdorff space	uncountable space, cardinality at least that of the continuum	connected and normal Hausdorff with at least two points implies cardinality at least that of the continuum

Opposite properties

• Totally disconnected space

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 148 (formal definition)
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 11 (formal definition)