Results 1 to 10 of 10

Math Help - interior sets

  1. #1
    Member
    Joined
    Feb 2010
    Posts
    133

    interior sets

    Greetings,

    I have two questions that I hope someone can assist me with. We let W_{1} and W_{2} be arbitrary subsets in the topological space M. I want to show that:

    1) int \left(W_{1} \cap W_{2} \right)=int \left(W_{2}\right) \cap int\left(W_{2}\right)

    2) int \left(W_{1} \cup W_{2} \right)\subseteq int \left(W_{2}\right) \cup int\left(W_{2}\right)

    I would greatly appreciate it if someone could help me through the problem so that I understand it and not just give me the answer.

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    (1) Let x be a point in the interior of W_1 \cap W_2. Then x is the center of an open ball B_1 fully contained in W_1 and a ball B_2 fully contained in W_2. Let B be the intersection of these two balls. B is in the interior of W_1 and the interior of W_2.

    Give number 2 a try yourself now.

    (By the way, I'm pretty sure you have 2 typos in the question - 2 of the W_2 should be W_1)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,918
    Thanks
    1762
    Awards
    1
    Remark: This is a topological space not necessarily metric space. It would be better to say open set in stead open ball.

    There is more wrong with #2 that just a typo.
    In space of real numbers with the usual topology let W_1=(0,1]~\&~W_2=(1,2).

    \text{Int} (W_1 ) = (0,1)\;,\;\text{Int} (W_2 ) = (1,2)\;\& \;\text{Int} (W_1  \cup W_2 ) = (0,2)
    Do you see something wrong now?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Reply to Plato at #3:

    I never took topology, so forgive me if I'm wrong. But isn't your W_{1} not in the usual topology on the reals? It's not an open set in the usual topology, right? So the theorem wouldn't apply?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,918
    Thanks
    1762
    Awards
    1
    Quote Originally Posted by Ackbeet View Post
    I never took topology, so forgive me if I'm wrong. But isn't your W_{1} not in the usual topology on the reals?
    This is a standard exercise in basic topology.

    There is nothing in the statement of the question to suggest that W_1\text{ and }W_2 are anything other than any two sets.

    The statement should be \text{Int}(W_1)\cup \text{Int}(W_2)\subseteq \text{Int}(W_1\cup W_2) .
    Follow Math Help Forum on Facebook and Google+

  6. #6
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Quote Originally Posted by Plato View Post
    There is nothing in the statement of the question to suggest that W_1\text{ and }W_2 are anything other than any two sets.
    So the specification in the OP that W_{1} and W_{2} are arbitrary subsets in the topological space M does not imply that they are open? I thought every subset in a topological space had to be open - indeed, was defined to be open.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,918
    Thanks
    1762
    Awards
    1
    Quote Originally Posted by Ackbeet View Post
    So the specification in the OP that W_{1} and W_{2} are arbitrary subsets in the topological space M does not imply that they are open? I thought every subset in a topological space had to be open - indeed, was defined to be open.
    Do not confuse these two:
    1. A subset of a topological space.
    2. A set in the topology of the space.

    The first means that an arbitrary set in the space and the second means that the set is in fact open (the set in the topology are open).
    Last edited by Plato; January 6th 2011 at 02:59 PM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    So, in looking at wiki's definition of a topological space, you're saying that your W_{1} and W_{2} are both arbitrary subsets of X, and not necessarily members of the topology \tau?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,918
    Thanks
    1762
    Awards
    1
    Quote Originally Posted by Ackbeet View Post
    So, in looking at wiki's definition of a topological space, you're saying that your W_{1} and W_{2} are both arbitrary subsets of X, and not necessarily members of the topology \tau?
    Well of course. Because an open set is its own interior then if W_1~\&~W_{2} were in the topology they would be open so there would be nothing to prove.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Got it. Thanks for letting me play the incompetent critic. I learned something!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Open sets and sets of interior points
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 9th 2011, 04:10 AM
  2. Clousre, Interior, Complement of sets
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: February 21st 2010, 12:53 AM
  3. Interior Sets
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 15th 2009, 12:28 PM
  4. Convex Sets - Interior and Closure
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 2nd 2009, 05:03 AM
  5. Interior, Boundary, and Closure of Sets
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 22nd 2009, 05:01 PM

Search Tags


/mathhelpforum @mathhelpforum