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
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1 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 (X 1, T 1 ) and (X 2, T 2 ) if X is the
2 disjoint union of X 1 and X 2 as sets and if T = {U 1 U 2 : U 1 T 1, U 2 T 2 }. In this case, X is disconnected, since X 1 and X 2 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
3 open intervals. (b) R n is connected. If R n 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,
4 hence non-empty Theorem. Let f be a continuous function from a connected topological space X to a topological space Y. Then f (X ) is connected Theorem. Let {E α } be a family of connected subsets of a topological space X such that E α E β for each pair α, β of indices. Then E α 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.
5 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) 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.
6 Example. Any countable metric space is totally disconnected. Hence R 0, 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} Theorem. Any interval in R is connected. Proof. Each open interval is connected because it is
7 Then X is connected, but not locally connected. 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 R 2 by X = {(x 1, 0) : x 1 R} {(x 1, x 2 ) : 0 x 2 1, x 1 rational}.
8 8.5. Theorem. Each connected component of a locally connected topological space is open.
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