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Connectedness. Connected, locally connected and disconnected sets and spaces. Components. Homotopic paths. Theorems.

Def. Disconnected set. A set which can be separated into two sets U and V which have no points in common and which are such that no accumulation point of U belongs to V and no accumulation point of V belongs to U. A set is totally disconnected if no subset containing more than one point is connected e.g. the set of rational numbers is totally disconnected..

James & James. Mathematics Dictionary

Def. Connected set. A set that cannot be separated into two sets U and V which have no points in common and which are such that no accumulation point of U belongs to V and no accumulation point of V belongs to U. The set of all rational numbers is not connected.

James & James. Mathematics Dictionary

Def. Locally connected set. A set S such that, for any point x of S and neighborhood U of x, there is a neighborhood V of x such that the intersection of S and V is connected and contained in U.

James & James. Mathematics Dictionary

Separated sets. Two subsets A and B of topological space X are said to be separated if 1) A and B are disjoint, and 2) neither contains an accumulation point of the other.

Thus two sets A and B are separated if and only if

Example 1. The subsets A = (4, 5) and B = (5, 7) are separated. The subsets C = (1, 3) and D = [3, 5) are not since D includes the accumulation point 3 of C.

Theorem 1. A set is connected if and only if it is not the union of two non-empty separated sets.

Def. Disconnected topological space. A topological space X that can be represented in the form X = A ∪ B, where A and B are two disjoint, non-empty open sets.

Def. Connected topological space. A topological space which is not disconnected.

Theorem 2. A topological space X is connected if and only if X and ∅ are the only subsets of X that are both open and closed.

Example of a disconnected space. The topological space X shown in Fig. 1 is disconnected. X consists of three closed, connected sets A, B and C in the plane. The sets A, B and C are not only closed but also open since X is both open and closed.

Theorem 3. A subset of the real line R that contains more than one point is connected if and only if it is an interval.

Theorem 4. Continuous images of connected sets are connected.

Theorem 5. The range of a continuous real function defined on a connected space is an interval.

Theorem 6. A topological space X is disconnected there exists a continuous mapping of X onto the discrete two-point space {0, 1}.

Theorem 7. The product of any non-empty collection of connected spaces is connected.

Theorem 8. The spaces Rn and Cn are connected.

Def. Maximal connected subset of a set. A maximal connected subset of a set S is a subset which is connected and is not a proper subset any connected subset of S.

Component of a topological space. A component E of a topological space X is a maximal connected subset of X.

Example 2. Fig. 1 shows a space X consisting of three components A, B and C.

Theorem 9. The components of a topological space X form a partition of X. In other words, the components are disjoint and their union is X. Every connected subset of X is contained in some component.

Theorem 10. If X is a arbitrary topological space, then:

1) each point in X is contained in exactly one component of X

2) each connected subspace of X is contained in a component

3) a connected subspace of X which is both open and closed is a component of X

4) each component of X is closed.

Considering 3) and 4) of this theorem, it is natural to ask the following question: Is a component of a topological space X also necessarily open? The answer is no. Consider the following example.

Example 3. Consider the set Q of all rational numbers. We note the following facts about Q.

1. Q is totally disconnected. This can be seen as follows: If x and z are any two distinct rational numbers, and if x < z, then there exists an irrational number y such that x < y < z. Consequently we can write

X = (- , y) ∪ (y, + )

which is a separation X. Thus Q is disconnected at every point y.

2. Because Q is totally disconnected, its components are its points.

3. The points (components) of Q are not open because any open subset of R which contains a given rational number also contains irrational numbers.

Def. Totally disconnected space. A topological space X is totally disconnected if no subset containing more than one point is connected

Examples of totally disconnected spaces.

1) discrete spaces

2) set of all rational numbers

3) set of all irrational numbers

4) Cantor set

Theorem 11. The components of a totally disconnected space are its points.

Theorem 12. Let X be a Hausdorff space. If X has an open base whose sets are also closed, then X is totally disconnected.

Theorem 13. Let X be a Hausdorff space. Then X is totally disconnected it has an open base whose sets are also closed.

Def. Locally connected space. A topological space X is said to be locally connected at a point p ε X if every open set containing p contains a connected open set containing p i.e. if the open connected sets containing p form a local base at p. A space X is said to be locally connected if it is locally connected at each of its points or, equivalently, if the open subsets of X form a base for X.

Example 4.

1. The spaces R, R2 and R3 are all locally connected.

2. Every discrete space X is locally connected. Why? For every p ε X, the set {p} is an open connected set containing p which is contained in every open set containing p (i.e. it meets the requirements of the above definition).

Def. Path. Let I = [0, 1], the closed unit interval. A path from a point a to a point b in a topological space X is a continuous function f: I → X with f(0) = a and f(1) = b. The point a is called the initial point of the path and b is called the terminal point.

Def. Arcwise connected set. A set such that each pair of its points can be joined by a curve all of whose points are in the set.

Syn. path connected set, pathwise connected set.

Theorem 14. Arcwise connected sets are connected.

The converse of this theorem is not true. A connected set is not necessarily arcwise connected as is illustrated by the following example.

Example 5. Consider the following subsets of the plane R2:

Here A consists of the points on the line segments joining the origin (0, 0) to the points (1, 1/n), n ε N as shown in blue in Fig. 2. B consists of the points on the x-axis between 1/2 and 1, shown in red. Now both sets A and B are arcwise connected. Thus both are also connected. In addition, A and B are not separated since each p ε B is a limit point of A. Consequently A ∪ B is connected. However, A ∪ B is not arcwise connected. There exists no path from any point in A to any point in B.

Homotopic paths. Given I = [0, 1], the closed unit interval, and a topological space X. Let f: I → X and g: I → X be two paths with the same initial point p ε X and the same terminal point q ε X. See Fig. 3b. Then f is said to be homotopic to g, written f g, if there exists a continuous function

H(s, t): I2 → X

such that

H(s, 0) = f(s)

H(s, 1) = g(s)

H(0, t) = p

H(1, t) = q

as indicated in the diagram of Fig. 3a. The function H(s, t) maps points labeled f (shown in red on the s axis) in Fig. 3a onto the path f in Fig. 3b in a one-to-one fashion. Similarly, it maps the points labeled g (shown in green) in Fig. 3a onto the path g in Fig. 3b in a one-to-one fashion. It also maps points labeled f1 and f2 of Fig. 3a onto the paths f1 and f2 of Fig. 3b.

The function H is called a homotopy from f to g. Two mappings f and g are homotopic if and only if they can be deformed into each other. A function H exists for two paths f and g if and only if the two paths can be continuously deformed into each other. Homotopic functions represent continuous deformations.

Example 6. In Fig. 4, the paths f1 and g1 in region R are not homotopic because they encircle a hole in the region and thus cannot be continuously deformed into each other (while staying in the region). The paths f2 and g2 are homotopic.

References

1. Lipschutz. General Topology

2. Simmons. Introduction to Topology and Modern Analysis

3. Munkres. Topology, A First Course

4. James & James. Mathematics Dictionary