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3 easy topology questions.

  1. Mar 19, 2012 #1
    My brain is giving me confusions.
    Which of these is true?
    1) Given a topology T and basis B, a set U is open iff for every x in U there exists basis element B with x belonging to B, and B contained in U.
    2) Given a topology T and basis B, a set U is open iff for every x in U there exists open set V with x belonging to V, and V contained in U.
    3) Given a topology T and basis B, a set U is open iff exery point of U belongs to the closure of U.
     
  2. jcsd
  3. Mar 19, 2012 #2
    The definition of open set that I was taught in topology* is that a set is open if it is the union of neighborhoods. How is this similar to your statement?

    This can most easily be answered by relating it to the "union of neighborhoods" definition, as a union of unions of neighborhoods is a union of neighborhoods. Thus an open set is a union of open sets.

    The definition I have learned for closure* is that the closure of a set is the set of all points near to the set. Is an element of the set near to that set?

    * I worded it this way in case different topology courses teach different (but equivalent) definitions for open sets.

    EDIT: Changed to follow homework help guidelines.
     
    Last edited: Mar 19, 2012
  4. Mar 20, 2012 #3
    1. "Given a topology T and basis B, a set U is open iff for every x in U there exists basis element b in B with x in b, and b a subset of U."

    For the "if", consider the definition of basis. For the "only if", recall that the elements of B are open, and consider what the definition of topological space says about unions of open sets. :smile:
     
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