§2.8. Connectedness a Topological Space X Is Said to Be Disconnected If X Is the Disjoint Union of Two Non-Empty Open Subsets

Home , Discrete space

§2.8. Connectedness A topological space X is said to be disconnected if X is the disjoint union of two non-empty open subsets. The space X is said to be connected if it is not disconnected. A subset of X is said to be a connected subset if it is connected in the relative topology. It is clear that X is disconnected iﬀ X has a closed open subset U such that U ≠ ∅ and U ≠ X . In this case, ∂U, the boundary of U, equals U \ int(U) = U \ U = ∅. We say a topological space (X , T ) is the disjoint union of two non-empty topological spaces (X1, T1) and (X2, T2) if X is the disjoint union of X1 and X2 as sets and if

T = {U1 ∪ U2 : U1 ∈ T1, U2 ∈ T2}.

In this case, X is disconnected, since X1 and X2 are two non-empty open sets. Conversely, suppose that X is disconnected. Then there are two disjoint non-empty open sets U and V such that X = U ∪ V . A subset W of X is open iff W ∩ U and W ∩ V are open. Hence X is the disjoint union of two non-empty topological spaces U and V with relative topologies. Therefore, X is disconnected iff X is expressible as the disjoint union of two non-empty topological spaces. Example. (a) R is connected: if U is an open set such that U ≠ R and U ≠ ∅, then ∂U ≠ ∅, since U is a disjoint union of open intervals. (b) Rn is connected. If Rn were the disjoint union of two non-empty open sets U and V , then there would be p ∈ U and q ∈ V , and the line X passing through p, q, which is homeomorphic to R, would be the disjoint union of two non-empty relatively open sets U ∩ X and V ∩ X . (c) A discrete space with more than one point is disconnected. (d) An uncountable space with cocountable topology is connected, since the intersection of any two non-empty open subsets is cocountable, hence non-empty. (e) An infinite space with cofinite topology is connected, since the intersection of any two non-empty open subsets is cofinite, hence non-empty. 8.1. Theorem. Let f be a continuous function from a connected topological space X to a topological space Y . Then f (X ) is connected.

8.2. Theorem. Let {Eα} be a family of connected subsets of a topological space X such that Eα ∩ Eβ ≠ ∅ for each pair

α, β of indices. Then ∪Eα is connected. Let X be a topological space and let x ∈ X . The connected component of x in X , denoted by C(x), is the union of all connected subsets of X that contain x. By Theorem 8.2, C(x) is connected. It is evidently the largest connected subset of X containing x. If E is a connected subset of X that meets C(x), then E ∪ C(x) is connected , so that it must be included in C(x). Hence C(x) includes each connected subset of X it meets. If C(x) meets C(y), then C(x) ⊂ C(y) and C(y) ⊂ C(x), and hence C(x) = C(y). 8.3. Theorem. Two connected components of X either coincide or are disjoint. The connected components of X form a partition of X into maximal connected subsets. 8.3a. Theorem. Each connected component of a topological space is closed. A topological space is said to be totally disconnected if the connected components are all singletons. Example. Any countable metric space is totally disconnected.

Hence R0, with inherited topology from R, is totally disconnected. Proof. Let X be a countable metric space with metric d. Fix x, y ∈ X , x ≠ y. Let a = d(x, y). Since the distances from x to other points form a countable set, there is a b in the interval (0, a) such that d(x, z) ≠ b for all z ∈ X . The sets {z : d(x, z) < b} and {z : d(x, z) > b} are a pair of complementary open sets separating x and y, hence y ̸∈ C(x). Therefore, C(x) = {x}. 8.4. Theorem. Any interval in R is connected. Proof. Each open interval is connected because it is homeomorphic to R. Suppose that I is a non-open interval and a is an end point of I . Let J = int(I ). Then J is an open interval and a belongs to the closure of J. Let x ∈ J. Then J ⊂ C(x), since J is connected. Since C(x) is closed, we see that a ∈ J ⊂ C(x). It follows that C(x) = I and I is connected. A topological space X is said to be locally connected if, for each p ∈ X and each open neighborhood U of p, there is a connected open neighborhood V of p with V ⊂ U. Example. Deﬁne a subset X of R2 by

X = {(x1, 0) : x1 ∈ R} ∪ {(x1, x2) : 0 ≤ x2 ≤ 1, x1 rational}.

Then X is connected, but not locally connected. 8.5. Theorem. Each connected component of a locally connected topological space is open.