Topology/Connectedness

Motivation

To best describe what is a connected space, we shall describe first what is a disconnected space. A disconnected space is a space that can be separated into two disjoint groups, or more formally:

A space $(X,𝒯)$ is said to be disconnected iff a pair of disjoint, non-empty open subsets $X_1,X_2$ exists, such that $X=X_1\cup X_2$.

A space $X$ that is not disconnected is said to be a connected space.

Examples

1. A closed interval [a,b] is connected. To show this, suppose that it was disconnected. Then there are two nonempty disjoint open sets A and B whose union is [a,b]. Let X be the the set equal to A or B and which contains x. Let s=sup{c|c∉X,c∈[a,b]}. If s is within X, then there is an open set (c-ε,c+ε) within X, implying that s is not an upper bound. If s is not within X, then s is within [a,b]\X, which is also open, and there is an open set (c-ε,c+ε) within [a,b]\X, implying that it is not the supremum.
2. The topological space $X=\{(0,1)\setminus \frac{1}{2}\}$ is disconnected: $A=(0,\frac{1}{2}),B=(\frac{1}{2},1)$
   A picture to illustrate:
   
   
   As you can see, the definition of a connected space is quite intuitive; when the space cannot be separated into (at least) two distinct subspaces.

Definition

A subset U of a topological space X is said to be clopen if it is both closed and open.

Theorems about connectedness

Theorem: The image of a connected set under a continuous function is also connected.
Proof: Let X be connected, and let f be a continuous function. Suppose f(X) is disconnected. Then there exists two nonempty disjoint sets A and B whose union is f(X). Then f^-1(A) and f^-1(A) are nonempty disjoint sets whose union is X, contradicting the fact that X is connected. Thus, f(X) is connected.

Note: this shows that connectedness is a topological property.

Theorem: If two connected sets have a nonempty intersection, then their union is connected.< br /> Proof: Suppose there are two open sets A and B which whose union is the union of X∪Y which are connected and have a nonempty intersection. Let x be in their intersection, and let suppose that x is C, which is either A or B. Let y be an element of Z, which is X or Y. If y is within the other set, then the intersection of the other set with Z and the intersection of C with Z makes two disjoint open nonempty sets who union is Z, contradicting the fact that Z is connected. Thus, y must be within the same set, implying that the other set is empty. Thus, there are no two nonempty open sets A and B which whose union is X∪Y, and so X∪Y is connected.

Exercises

• For any topological space X, the empty set and the entire space are clopen.

1. Show that a topological space X is disconnected if and only if it has clopen sets other than $\mathrm{\varnothing }$ and $X$ (Hint: Why is $X_1$ clopen?)
2. Prove the mean value theorem: if f:R→R is a continuous function for [a,b], then for any y between f(a) and f(b), there exists a c∈[a,b] such that f(c)=y.
3. Prove that R is not homeomorphic to R² (hint: removing a single point from R makes it disconnected).