Connected space
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.
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Definition
Symbol-free definition
A topological space is said to be connected if it satisfies the following equivalent conditions:
- It cannot be expressed as a disjoint union of two nonempty open subsets
- It cannot be expressed as a disjoint union of two nonempty closed subsets
- It has no clopen subsets other than the empty subspace and the whole space
Relation with other properties
Stronger properties
| property | quick description | 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 |
Opposite properties
Metaproperties
Products
This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products
Any product of connected spaces is connected in the product topology (this is true for finite, as well as infinite, products). For full proof, refer: Connectedness is product-closed
Box products
This property of topological spaces is a box product-closed property of topological spaces: it is closed under taking arbitrary box products
View other box product-closed properties of topological spaces
A box product of connected spaces is connected.
Coarsening
This property of topological spaces is preserved under coarsening, viz, if a set with a given topology has the property, the same set with a coarser topology also has the property
Switching to a coarser topology continues to keep the topological space connected. This is because if there is a proper nonempty clopen subset in the coarser topology, it would also be there in the finer topology. Template:Connected union-closed
If a topological space is the union of a family of subsets, with the property that they all have a nonempty intersection, then the whole space is connected.
A slight variant is this: if a topological space is the union of a subspace and a family of other subspaces that each intersect this subspace nontrivially, and all these subspaces are connected, then the whole space is connected.
Closure under continuous images
The image, via a continuous map, of a topological space having this property, also has this property
If is a connected space and is the image of via a continuous map, then is also connected. For full proof, refer: connectedness is continuous image-closed
References
Textbook references
- Topology (2nd edition) by James R. MunkresMore info, Page 148 (formal definition)
- Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. ThorpeMore info, Page 11 (formal definition)