Topology/Connectedness

Motivation

Template:TOC right 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 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 $(s-\epsilon ,s+\epsilon )$ 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 $(s-\epsilon ,s+\epsilon )$ 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: $A=(0,\frac{1}{2}),B=(\frac{1}{2},1)$
   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.

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, $a\in X\mathrm{\forall }a\in A$ ...(1).

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

Hence, similarly, $b\in Y\mathrm{\forall }b\in B$ ...(2)
Now, consider $c\in A\cap B$. From (1) and (2), $c\in X\cap Y$, and hence $X\cap Y\ne \mathrm{\varnothing }$. As $X,Y\in 𝒯$ 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 $\mathrm{\varnothing }$ 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 $f:[a,b]\to ℝ$ 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 $ℝ$ is not homeomorphic to ${ℝ}^2$ (hint: removing a single point from $ℝ$ 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)

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