Connected space
A topological space that cannot be split into two disjoint open subsets.
In topology, a connected space is one that cannot be split into two or more disjoint, non-empty open pieces. This property is a fundamental way to tell topological spaces apart. A subset of a space is called a connected set if it itself is a connected space when given the subspace topology. A space is disconnected if it can be written as the union of two disjoint, non-empty open sets; otherwise, it is connected. (Some authors exclude the empty set from being connected, but that convention is not followed here.) The modern definition, based on the idea that a space cannot be split into two separated sets, was developed independently by N.J. Lennes, Frigyes Riesz, and Felix Hausdorff around the start of the 20th century. Connectedness creates an equivalence relation: two points are equivalent if they lie in the same connected subset. The connected component of a point is the largest connected subset containing that point—it is the union of all connected subsets that include it. The connected components of a space are its maximal connected subsets (ordered by inclusion). They partition the space: they are disjoint, non-empty, and their union is the whole space. In fact, a component is exactly an equivalence class under the relation described above. Every component is a closed subset of the original space. If there are only finitely many components, each is also open. But with infinitely many components, this may not hold. For example, the connected components of the rational numbers are single points (singletons), which are not open. To see this, take two distinct rationals q₁ < q₂. Choose an irrational number r between them. Then the sets A = {q ∈ ℚ : q < r} and B = {q ∈ ℚ : q > r} form a separation of ℚ, with q₁ in A and q₂ in B. Thus no two distinct rationals can be in the same component, so each component is a one-point set.
- Field
- Topology
- Known for
- Definition of connectedness, connected components, and related properties
- Related concepts
- Path connected, simply connected, n-connected, locally connected, totally disconnected, totally separated
Lore & Background
The modern formulation of connectedness—in terms of no partition of the space into two separated sets—first appeared independently with N.J. Lennes, Frigyes Riesz, and Felix Hausdorff at the beginning of the 20th century. Connectedness defines an equivalence relation: two points are equivalent if they belong to the same connected subset. The connected component of a point is the union of all connected subsets containing that point, forming the unique largest connected subset containing it. The connected components of a non-empty topological space form a partition: they are disjoint, non-empty, and their union is the whole space. Every component is a closed subset; if the number of components is finite, each is also open, but this may not hold for infinite components, as seen with the rational numbers, where each component is a singleton that is not open.
Reader's Guide
Connectedness is a fundamental topological property that distinguishes spaces based on whether they can be separated into disjoint open parts. The concept underpins many areas of topology and analysis, providing a basis for understanding continuity, paths, and separation. The source article notes that related but stronger conditions include path connected, simply connected, and n-connected, while locally connected neither implies nor follows from connectedness. The notion of connected components allows the decomposition of any topological space into maximal connected pieces, which are always closed. The article also discusses totally disconnected spaces (where all components are one-point sets) and the stricter notion of totally separated spaces, giving an example of a totally disconnected space that is not totally separated. The historical development by Lennes, Riesz, and Hausdorff established the modern definition, which remains central to topology.
Did You Know?
- The connected components of the rational numbers are one-point sets (singletons), which are not open.
- A space is totally disconnected if all its connected components are one-point sets.
Frequently Asked Questions
What exactly is a connected space in topology?
A connected space is a topological space that cannot be broken apart into two non-empty, disjoint open subsets. It is one of the most basic ways topologists distinguish one space from another, sitting right at the foundation of the subject.
How does connectedness differ from path connectedness?
A space can be connected without being path connected, meaning you cannot always draw a continuous curve between any two points even though the space cannot be split into two open pieces. Path connectedness is a strictly stronger condition that implies connectedness, but the reverse does not hold.
What are connected components and why do they matter?
Connected components are the maximal connected subsets of a space, essentially the largest indivisible chunks that remain connected. Every topological space decomposes uniquely into its connected components, making them a fundamental building block for understanding the space's overall structure.
Is the empty set considered connected?
Under the standard definition used here, the empty set is treated as connected because it simply cannot be expressed as a union of two non-empty disjoint open sets. Some authors adopt the opposite convention, but that stricter reading is not followed in this entry.
How does a connected space relate to concepts like simply connected or n-connected?
Simply connected and n-connected are stronger, more specific conditions that layer additional requirements about loops and higher-dimensional holes on top of basic connectedness. A space must first be connected before questions about simple connectivity or higher connectivity even make sense.
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