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Math and science::Topology

Connectedness. Equivalent formulations

For a nonempty topological space \( X \), the following are equivalent:

  1. \( X \) is connected;
  2. the only subsets that are both open and closed are \( \emptyset \) and \( X \);
  3. every continuous map from \( X \) to a discrete space is constant;
  4. every continuous map from \( X \) to the two-point discrete space is constant.

The 3rd and 4th statement are similar, yet both useful in their own right: the 3rd statement is general and allows progress to be made in a proof when one knows that \( X \) is connected. The 4th is powerful in the reverse direction: simply proving that all continuous maps to the two-point discrete space is constant is enough to show that \( X \) is connected.

Context

Source

Leinster, p 69