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 $$(X,\mathcal{T})$$ 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
A picture to illustrate:
 * 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 set equal to $$A$$ or $$B$$ and which does not contain $$b$$. Let $$s=\sup X$$. Since X does not contain b, s must be within the interval [a,b] and thus must be within either X or $$[a,b]\setminus X$$. If $$s$$ is within $$X$$, then there is an open set $$(s-\varepsilon,s+\varepsilon)$$ within $$X$$. If $$s$$ is not within $$X$$, then $$s$$ is within $$[a,b]\setminus X$$, which is also open, and there is an open set $$(s-\varepsilon,s+\varepsilon)$$ within $$[a,b]\setminus X$$. Either case implies that $$s$$ is not the supremum.
 * 2) The topological space $$X = (0,1)\setminus\{{\frac{1}{2} } \}$$ is disconnected: $$A = (0,\frac{1}{2} ), B = (\frac{1}{2}, 1)$$

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.

Definitions
Definition 1.1

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

Definition 1.2

A topological space X is said to be totally disconnected if every subset of X having more than one point is disconnected under the subspace topology

Theorems about connectedness
If $$X$$ and $$Y$$ are homeomorphic spaces and if $$X$$ is connected, then $$Y$$ is also connected.

Proof: Let $$X$$ be connected, and let $$f$$ be a homeomorphism. Assume that $$Y$$ is disconnected. Then there exists two nonempty disjoint open sets $$Y_1$$ and $$Y_2$$ whose union is $$Y$$. As $$f$$ is continuous, $$f^{-1}(Y_1)$$ and $$f^{-1}(Y_2)$$ are open. As $$f$$ is surjective, they are nonempty and they are disjoint since $$Y_1$$ and $$Y_2$$ are disjoint. Moreover, $$f^{-1}(Y_1) \cup f^{-1}(Y_2)=f^{-1}(Y)=X$$, contradicting the fact that $$X$$ is connected. Thus, $$Y=f(X)$$ is connected.

Note: this shows that connectedness is a topological property.

If two connected sets have a nonempty intersection, then their union is connected.

Proof: Let $$A$$ and $$B$$ be two non-disjoint, connected sets. Let $$X$$ and $$Y$$ be non-empty open sets such that $$X\cup Y=A\cup B$$. Let $$a_0\in A$$. Without loss of generality, assume $$a_0\in X$$.

As $$A$$ is connected, $$a\in X\forall a\in A$$ ...(1).

As $$Y$$ is non-empty, $$\exists b\in B$$ such that $$b\in Y$$.

Hence, similarly, $$b\in Y\forall b\in B$$ ...(2) Now, consider $$c\in A\cap B$$. From (1) and (2), $$c\in X\cap Y$$, and hence $$X\cap Y\neq\emptyset$$. As $$X,Y\in\mathcal{T}$$ are arbitrary, $$A\cup B$$ is connected.

If two topological spaces are connected, then their product space is also connected.''

Proof: Let X1 and X2 be two connected spaces. Suppose that there are two nonempty open disjoint sets A and B whose union is X1×X2. If for every x∈X, {x}×X2 is either completely within A or within B, then π1(A) and π1(B) are also open, and are thus disjoint and nonempty, whose union is X1, contradicting the fact that X1 is connected. Thus, there is an x∈X such that {x}×X2 contains elements of both A and B. Then π2(A∩{(x,y)}) and π2(B∩{(x,y)}), where y is any element of X2, are nonempty disjoint sets whose union is X2, and which are a union of open sets in {(x,y)} (by the definition of product topology), and are thus open. This implies that X2 is disconnected, a contradiction. Thus, X1×X2 is connected.

Exercises

 * 1) Show that a topological space $$X$$ is disconnected if and only if it has clopen sets other than $$\emptyset$$ and $$X$$ (Hint: Why is $$X_1$$ clopen?)
 * 2) Prove that if $$f:X\to Y$$ is continuous and surjective (not necessarily homeomorphic), and if $$X$$ is connected, then $$Y$$ is connected.
 * 3) Prove the Intermediate Value Theorem: if $$f:[a,b]\to \mathbb{R}$$ is continuous, then for any $$y$$ between $$f(a)$$ and $$f(b)$$, there exists a $$c\in [a,b]$$ such that $$f(c)=y$$.
 * 4) Prove that $$\mathbb{R}$$ is not homeomorphic to $$\mathbb{R}^2$$ (hint: removing a single point from $$\mathbb{R}$$ makes it disconnected).
 * 5) Prove that an uncountable set given the countable complement topology is connected (this space is what mathematicians call 'hyperconnected')
 * 6) a)Prove that the discrete topology on a set X is totally disconnected.

b) Does the converse of a) hold (Hint: Even if the subspace topology on a subset of X is the discrete topology, this need not imply that the set has the discrete topology)