In topology, a connected space is a topological space that cannot be represented as the union of two non‑empty, disjoint, open subsets. Equivalently, a space $X$ is connected if the only subsets of $X$ that are both open and closed (clopen) are the empty set $\varnothing$ and $X$ itself.
Formal Definition
Let $(X,\tau)$ be a topological space, where $\tau$ denotes the collection of open sets. $X$ is connected if there do not exist open sets $U,V \in \tau$ such that:
- $U eq \varnothing$ and $V eq \varnothing$;
- $U \cap V = \varnothing$;
- $U \cup V = X$.
If such a pair $(U,V)$ exists, the space is called disconnected, and $(U,V)$ is a separation of $X$.
Equivalent Characterizations
- The only clopen subsets of $X$ are $\varnothing$ and $X$.
- Every continuous map $f : X \to {0,1}$ (where ${0,1}$ carries the discrete topology) is constant.
- Any two points of $X$ cannot be separated by a pair of disjoint open neighborhoods whose closures are also disjoint.
Examples
| Space | Connected? | Reason |
|---|---|---|
| $\mathbb{R}$ with the usual topology | Yes | No non‑trivial clopen subsets; intervals are connected. |
| $[0,1]$ with the subspace topology of $\mathbb{R}$ | Yes | Closed interval is a continuum; cannot be split into disjoint open subsets in the subspace topology. |
| $\mathbb{R} \setminus {0}$ | No | Can be separated into $(-∞,0)$ and $(0,∞)$. |
| Finite discrete space with more than one point | No | Every singleton is open; any two distinct points give a separation. |
| The topologist’s sine curve (with the subspace topology of $\mathbb{R}^2$) | Yes | Though highly pathological, it is connected but not path‑connected. |
Properties
- Continuous Images: The continuous image of a connected space is connected. If $f : X \to Y$ is continuous and $X$ is connected, then $f(X)$ is connected in $Y$.
- Products: Arbitrary products of connected spaces are connected (Tychonoff’s theorem). Finite products preserve connectedness directly.
- Subspaces: A subspace of a connected space need not be connected; however, any connected subspace that contains a point and the closure of a connected set containing that point is also connected.
- Path‑Connectedness: Every path‑connected space is connected, but the converse is not generally true.
- Components: The maximal connected subsets of a space are called connected components; they form a partition of the space. Each component is closed, though not necessarily open.
Related Concepts
- Locally Connected: A space where every point has a basis of connected neighborhoods.
- Totally Disconnected: A space whose only connected subsets are singletons.
- Arc‑Connected: A stronger form of path‑connectedness requiring arcs (homeomorphic images of $[0,1]$).
- Separation Axioms: Connectedness interacts with separation properties (e.g., a connected Hausdorff space cannot be partitioned by clopen sets).
References
- Munkres, James R. Topology (2nd ed.). Prentice Hall, 2000.
- Willard, Stephen. General Topology. Addison‑Wesley, 1970.
- Steen, Lynn Arthur; Seebach, J. Arthur, Counterexamples in Topology, Springer, 1978.
These sources provide formal definitions, proofs of the listed properties, and numerous examples illustrating the concept of a connected space in general topology.
