Connected space

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

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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

• Path-connected space
• Simply connected space

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 (this is true for finite, as well as infinite, products).

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 $X$ is a connected space and $Y$ is the image of $X$ via a continuous map, then $Y$ is also connected.