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 are both open and closed.
Lemma. Union and intersection of open sets
Let
- Let
be a family (finite or not) of open subsets of . Then is also open in X. -
Let
and be open subsets of . Then is also open in X. - Special cases
and are open also.
The mirror statements for closed sets are as follows.
Lemma. Union and intersection of closed subsets
Let
-
Let
and be closed subsets of . Then is also closed in X. -
Let
be a family (finite or not) of closed subsets of . Then is also open in X.
Special cases
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Context
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