Skip to content

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