Topology MOC

Connectedness

A topological space (𝑋,T) is called connected¹ if the following equivalent² conditions hold: topology

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

Proof of equivalence of definitions

For any function 𝑓 :𝑋 →{0,1}, it follows that 𝑓−1{0} ∩𝑓−1{1} =∅ and 𝑓−1{0} ∪𝑓−1{1} =𝑋. Clearly 𝑓−1∅ 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 ∈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

• Main theorem: The continuous image of a connected space is connected
• Connected fibres and quotient implies connected space
• Connectedness is transitive
• Connected subspaces of the real line are intervals
• Connectedness is homotopy invariant
• Cut point

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Footnotes

1. German zusammenhängend. Connectedeness is Zusammenhang. ↩

2. The footnotes indicate which is the primary definition for a given source. ↩

3. 2020, Topology: A categorical approach, p. 39 ↩

4. 2010, Algebraische Topologie, p. 15 ↩