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 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 such that are both nonempty, is empty, and . |
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 such that are both nonempty, is empty, and . |
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 as a union where are all nonempty and open, is empty for , and 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 as a union where are all nonempty and closed, is empty for , and . |
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 such that is clopen, i.e., is both open and closed, either is empty or . |
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 and a subset of such that 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 and a closed subset of such that is not connected in the subspace topology. |
product-closed property of topological spaces | Yes | connectedness is product-closed | Suppose , are all connected spaces. Then, the Cartesian product 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 all connected spaces such that the Cartesian product is not connected in the box topology. |
coarsening-preserved property of topological spaces | Yes | connectedness is coarsening-preserved | If is connected under a topology , it remains connected when we pass to a coarser topology than . |
continuous image-closed property of topological spaces | Yes | connectedness is continuous image-closed | If is a connected space and is the image of under a continuous map, then 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 is a subset of that is connected in the subspace topology. Then, the closure 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 spacesView 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.
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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
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)