Connectedness

Definition: Connectedness of a Topological Space

A topological space is connected iff it cannot be represented as the union of two disjoint, nonempty open sets.

Definition: Connectedness of a Subset

Let $(X,{\tau }_X)$ be a topological space.

A subset $S\subseteq X$ is connected iff it is connected as a subspace.

Definition: Disconnectedness

A topological space is disconnected iff it is not connected.

Connectedness Criteria

THEOREM

A topological space $(X,\tau )$ is connected if and only if its only clopen subsets are $\varnothing$ and $X$.

PROOF

We need to prove two things:

• (I) If $(X,\tau )$ is connected, then its only clopen subsets are $\varnothing$ and $X$.
• (II) If the only clopen subsets of $(X,\tau )$ are $\varnothing$ and $X$, then $(X,\tau )$ is connected.

Proof of (I):

We prove this by contradiction.

Suppose that $(X,\tau )$ is connected and let $U\subset X$ be clopen. If $U$ is nonempty, then so is $X\setminus U$. Since $U$ is clopen, so is its complement $X\setminus U$. More importantly, this implies that both $U$ and $X\setminus U$ are open. However, since $X=U\cup X\setminus U$, this means that $X$ can be represented as the union of two disjoint, nonempty open sets and is thus not disconnected, which is a contradiction.

Proof of (II):

We prove this by contradiction.

Suppose that the only clopen subsets of $(X,\tau )$ are $\varnothing$ and $X$. Assume that $(X,\tau )$ is disconnected, i.e. there exists two disjoint, nonempty open sets $U$ and $V$ such that $X=U\cup V$.

THEOREM

Let $(X,\tau )$ be a topological space and let $S$ be a subset of $X$.

If $S$ can be expressed as the union of a collection $\mathcal{C}$ of subsets whose intersection is nonempty, then $S$ is also .

PROOF

TODO