An exercise on topologies from closure operators

/ Christian Nguembou Tagne

It is well-known that a topology on a set can be defined by the means of a neighborhood system. Topologies can also arise from certain operators. In this note, we set an exercise, which invites to prove that topologies can be obtained from closure operators. The exercise statement below provides a clear definition of closure operators.

Exercise.

Let X be a set and let M\mapsto\overline{M} be a mapping of the power set \mathcal{P}(X) onto itself such that:

  1. \overline{\emptyset}=\emptyset;
  2. for every M\subset X, we have M\subset\overline{M};
  3. for every M\subset X, we have \overline{\overline{M}}=\overline{M};
  4. for all M\subset X and N\subset X, we have \overline{M\cup N}=\overline{M}\cup\overline{N}.

Show that there is a unique topology on X such that \overline{M} is the closure of M with respect to this topology, for all M\subset X (Hint. define the topology by means of its closed sets).

Exercise as PDF file

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Une réflexion sur “An exercise on topologies from closure operators”

  1. Pingback: Topologies from closure operators | Formalis Mathematica

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