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Math and science::Topology

Metric space. Unions and intersections of open sets

  • An arbitrary union of open subsets is open.
  • A finite intersection of open subsets is open.
  • Special cases: and X are both open and closed.

Lemma. Union and intersection of open sets

Let X be a metric space.

  • Let (Ui)iI be a family (finite or not) of open subsets of X. Then iIUi is also open in X.
  • Let U1 and U2 be open subsets of X. Then U1U2 is also open in X.
  • Special cases and X are open also.


The mirror statements for closed sets are as follows.

Lemma. Union and intersection of closed subsets

Let X be a metric space.

  • Let V1 and V2 be closed subsets of X. Then V1V2 is also closed in X.
  • Let (Vi)iI be a family (finite or not) of closed subsets of X. Then iIVi is also open in X.

Special cases and X are also closed.


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