A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
A topological space is connected if it cannot be written as a union of two disjoint nonempty open sets. Equivalently, the only subsets that are both open and closed (clopen) are the empty set and the whole space. Connectedness captures the intuitive idea that a space is "in one piece." The real line ℝ is connected, but ℝ minus a point is not—removing any point splits it into two open rays. The continuous image of a connected space is connected, which is why the intermediate value theorem holds: a continuous function on a connected domain cannot skip values. Connectedness is a topological invariant preserved under homeomorphisms.
Prove that ℝ is connected using the least upper bound property, then show ℚ is disconnected by exhibiting a clopen set. Working through these two cases builds a concrete understanding of the definition before moving to more exotic spaces.
Connected does not mean path-connected. The topologist's sine curve is connected but not path-connected. Students also sometimes think removing a point always disconnects a space—this is true for ℝ but false for ℝ² (which remains connected after removing any single point).
A topological space X is connected if it cannot be written as the union of two disjoint nonempty open sets. Equivalently, the only subsets of X that are both open and closed (clopen) are ∅ and X itself. If such a nontrivial partition X = U ∪ V with U, V disjoint, nonempty, and open does exist, then X is disconnected, and the pair (U, V) is called a separation or disconnection of X. Connectedness captures the intuition that the space is "in one piece" — there is no way to split it into two topologically isolated halves.
The real line ℝ with the standard topology is connected. The proof uses the least upper bound property: if ℝ = U ∪ V were a separation, take a point a ∈ U and b ∈ V with a < b, and consider s = sup(U ∩ [a, b]). Then s must belong to one of U or V, and in either case a contradiction arises because both U and V are open (an open set around s must extend past the boundary). This argument is essentially why the intermediate value theorem holds: continuous functions on connected spaces cannot "skip" values. If f : [a, b] → ℝ is continuous with f(a) < c < f(b), and f never equals c, then the preimages f⁻¹((−∞, c)) and f⁻¹((c, ∞)) would form a separation of [a, b], contradicting connectedness.
The rational numbers ℚ provide the canonical example of a disconnected space. Since √2 is irrational, the sets U = {q ∈ ℚ : q < √2} and V = {q ∈ ℚ : q > √2} are disjoint, nonempty, and together cover all of ℚ. Both are open in the subspace topology (each is the intersection of ℚ with an open ray in ℝ). So (U, V) is a separation, and ℚ is disconnected. The "hole" at √2 is what allows the split. This illustrates a general pattern: missing points can destroy connectedness. Removing a single point from ℝ disconnects it into two open rays, but removing a single point from ℝ² does not disconnect it — in two dimensions, paths can detour around the missing point.
Connectedness is preserved by continuous maps: if f : X → Y is continuous and X is connected, then f(X) is connected. This is a powerful and broadly applicable principle. It means that topological invariants defined via connectedness (such as the number of connected components) are preserved by homeomorphisms. It also provides a method for proving connectedness: to show a space is connected, exhibit it as the continuous image of a known connected space. Conversely, to show two spaces are not homeomorphic, show they have different numbers of connected components — or that removing a point disconnects one but not the other.
Connected does not imply path-connected. The topologist's sine curve — the closure of {(x, sin(1/x)) : x > 0} in ℝ² — is connected but not path-connected. No continuous path can traverse the infinitely oscillating accumulation near x = 0. In locally path-connected spaces (including most spaces arising in geometry and analysis), connectedness and path-connectedness coincide, but in general topology they are distinct concepts, with path-connectedness being strictly stronger.