Topologies from closure operators

/ Christian Nguembou Tagne

Topologies on a set can be defined by the means of neighborhood systems. In this note, we prove that topologies can also be constructed using closure operators. In fact, the note is a solution of an exercise formulated in a previous entry of this blog. We repeat the statement of the exercise in question here, as a reminder.

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).

Solution of the exercise as PDF file

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2 réflexions sur “Topologies from closure operators”

  1. Pingback: An exercise on topologies from interior operators | Formalis Mathematica

  2. Pingback: Topologies from interior operators | Formalis Mathematica

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