Connectedness

Definition: Connectedness of a Topological Space

A topological space is connected if it cannot be represented as the union of two disjoint, non-empty open sets.

Definition: Connectedness of a Subset

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

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

Definition: Disconnectedness

A topological space is disconnected if it is not connected.

Theorem: Connectedness and Clopen Sets

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

PROOF

TODO

Theorem: Connectedness of Subsets

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 connected subsets with a non-empty intersection, then $S$ is also connected.

PROOF

TODO