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Noetherian Topological Spaces

Posted: Sat Dec 19, 2015 10:53 am
by Tsakanikas Nickos
  1. Let \( X \) be a Noetherian topological space. Show that every subspace of \( X \) is also Noetherian and that \( X \) quasi-compact.
  2. Let \( X \) be a topological space. Show that the following are equivalent:
    1. \( X \) is Noetherian.
    2. Every open subspace of \( Χ \) is quasi-compact.
    3. Every subspace of \( Χ \) is quasi-compact.

Note: "Quasi-compact" means that every open cover of \( X \) has a finite subcover.