Connectedness

A topological space $(𝑋, T)$ is called connected¹ if the following equivalent² conditions hold: #m/def/topology

any continuous function $𝑓 : 𝑋 \to {0, 1}$ with discrete codomain is constant.³
$𝑋$ is not the union of two non-empty disjoint open sets.⁴
the only clopen sets are $\emptyset$ and $𝑋$ .

Proof of equivalence of definitions

For any function $𝑓 : 𝑋 \to {0, 1}$ , it follows that $𝑓 - 1 {0} \cap 𝑓 - 1 {1} = \emptyset$ and $𝑓 - 1 {0} \cup 𝑓 - 1 {1} = 𝑋$ . Clearly $𝑓 - 1 \emptyset$ and $𝑓 - 1 {0, 1}$ are open, so $𝑓$ is continuous iff $𝑓 - 1 {0}$ and $𝑓 - 1 {1}$ are open. Hence condition 1 and 2 are equivalent. A partition of the space into two open subsets implies both of those subsets are clopen, and likewise if we have a clopen subset the space can be partitioned into it and its likewise clopen compliment, thus condition 3 is equivalent to 1 and 2.

A stronger property is path-connected. When a subset is said to be connected it is meant under the Subspace topology.

Connected components

Two points $𝑥, 𝑦 \in 𝑋$ are said to be connected iff there exists a connected subspace containing both points. This is an equivalence relation (Connectedness is transitive) and the equivalence classes are called connected components of $𝑋$ .

Properties

#state/tidy | #lang/en | #SemBr

German zusammenhängend. Connectedeness is Zusammenhang. ↩
The footnotes indicate which is the primary definition for a given source. ↩
2020, Topology: A categorical approach, p. 39 ↩
2010, Algebraische Topologie, p. 15 ↩

Connectedness

Connected components

Properties

Footnotes