Results 1 to 2 of 2

Math Help - closure

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    156

    closure

    Show that  X- E^{\circ} = \overline{X-E} where X is a metric space and E is a subset of X. Can we do this without using DMorgan's Laws (e.g. without using intersection of closed sets)?

    Because you are taking away  E^{\circ} \subset E from  X . Whereas  \overline{X-E} = X-E \cup (X-E)' where  (X-E)' is the set of limit points of  X-E . So you are taking away more points from  X . Adding back the limit points gives you  X- E^{\circ} (e.g. you are not adding any interior points).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jan 2006
    From
    Gdansk, Poland
    Posts
    117
    First of all we should notice two facts:

    1.
    X \setminus  E^\circ \supset X \setminus E

    2.
    X \setminus \overline{X \setminus  E} \subset (X \setminus X \setminus E) = E

    Ad 1.
    It follows from E^\circ \subset E

    Ad 2.
    It follows from X \setminus E \subset \overline{X \setminus E}

    Now, since E^\circ is open, X \setminus E^\circ is closed.
    Since closure of A is the smallest closed set containing A from 1 we have:
    X \setminus  E^\circ \supset \overline{X \setminus E}

    Now, since \overline{X \setminus  E} is closed, X \setminus \overline{X \setminus  E} is open.
    Since interior of A is the largest open set in A we from 2 we have

    X \setminus \overline{X \setminus  E} \subset E^\circ

    Thus: \overline{X \setminus  E} \supset X \setminus E^\circ

    So we have: \overline{X \setminus  E} = X \setminus E^\circ
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Relation between topological closure and algebraic closure
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 4th 2010, 02:45 PM
  2. closure
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: March 27th 2010, 12:01 AM
  3. closure
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 27th 2009, 01:03 PM
  4. Replies: 6
    Last Post: February 11th 2009, 12:56 PM
  5. Closure
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 6th 2009, 09:06 PM

Search Tags


/mathhelpforum @mathhelpforum