Results 1 to 4 of 4

Math Help - Is the sum of two compact subsets of Rn compact + another problem

  1. #1
    Newbie BBAmp's Avatar
    Joined
    May 2010
    From
    Cleveland,OH/Madison,WI/Queens,NY
    Posts
    17

    Is the sum of two compact subsets of Rn compact + another problem

    Hello everyone,

    1) I have been banging my head against the wall trying to figure out how to prove that two subsets A, B which are compact subsets of R^n produce another compact subset when added together.

    The problem is.. I know the definition of a compact set, but is that the right place to start? Can anyone please gently guide me in the right direction?

    2) if U is an open subset in Rn and V is an arbitrary subset, then U+V is open. Here I am having issues because again I know the definition of an open subset but I become paralyzed when I try to do anything with it. Frankly I'm not sure if V is an arbitrary subset of Rn but even if I assume it is, how on earth do I start?

    I am still very new to proofs and I would like to learn on my own so please post only ideas of where I can or should start.

    Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie BBAmp's Avatar
    Joined
    May 2010
    From
    Cleveland,OH/Madison,WI/Queens,NY
    Posts
    17
    okay this is how i went about solving the first one.. I would really appreciate input on if my thought process is correct since I am learning how to write proofs.

    since A and B are compact subsets of Rn, A and B each have open covers where (O_{\alpha})_{\alpha\epsilon scriptA} is an open cover of A (where scriptA is some index set), and (P_{\beta})_{\beta\epsilon scriptB} is an open cover of B (where scriptB is another index set). Then there are \alpha_{1},...,\alpha_{m} \epsilon scriptA and \beta_{1},...,\beta_{m} \epsilon scriptB such that A \subseteq \cup O_{\alpha} and B \subseteq \cup P_{\beta} which comes from the definition of compactness.

    Adding A and B creates a new set C. In order for C to be compact, it must have its own set of open covers where the union of a certain collection of them (much like the union of open covers that contain A and B above) where C \subseteq \cup Q_{\gamma}. That means:

    C = (A+B) \subseteq ( \cup O_{\alpha} + \cup P_{\beta})

    then since  C = A+B \Rightarrow C \subseteq (\cup O_{\alpha} + \cup P_{\beta})
    also since  C = A+B \Rightarrow A+B \subseteq (\cup Q_{\gamma})

    which implies \cup O_{\alpha} + \cup P_{\beta} = \cup Q_{\gamma} so c is compact.

    Input please! Thanks again in advance.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,307
    Thanks
    1284
    Quote Originally Posted by BBAmp View Post
    okay this is how i went about solving the first one.. I would really appreciate input on if my thought process is correct since I am learning how to write proofs.

    since A and B are compact subsets of Rn, A and B each have open covers where (O_{\alpha})_{\alpha\epsilon scriptA} is an open cover of A (where scriptA is some index set), and (P_{\beta})_{\beta\epsilon scriptB} is an open cover of B (where scriptB is another index set). Then there are \alpha_{1},...,\alpha_{m} \epsilon scriptA and \beta_{1},...,\beta_{m} \epsilon scriptB such that A \subseteq \cup O_{\alpha} and B \subseteq \cup P_{\beta} which comes from the definition of compactness.
    This is an incorrect definition of "compactness". Saying that a set is compact does NOT mean that there exist open covers that have finite subcovers- it means that every open cover has a finite subcover. In particular, that means that you cannot assume that an open cover of A+ B is constructed from given open covers of A and B.

    By the way, how are you defining A+ B for A and B in \mathbb{R}^n? \{u+ v| u\in A, v\in B\} and "a+ b" is coordinatewise addition?

    Adding A and B creates a new set C. In order for C to be compact, it must have its own set of open covers where the union of a certain collection of them (much like the union of open covers that contain A and B above) where C \subseteq \cup Q_{\gamma}. That means:

    C = (A+B) \subseteq ( \cup O_{\alpha} + \cup P_{\beta})

    then since  C = A+B \Rightarrow C \subseteq (\cup O_{\alpha} + \cup P_{\beta})
    also since  C = A+B \Rightarrow A+B \subseteq (\cup Q_{\gamma})

    which implies \cup O_{\alpha} + \cup P_{\beta} = \cup Q_{\gamma} so c is compact.

    Input please! Thanks again in advance.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Oct 2012
    From
    India
    Posts
    1

    Re: Is the sum of two compact subsets of Rn compact + another problem

    take two sets A and B then define f(x,y)=x+y, x belongs to A and y belongs to B, f is continuous, continuous image of compact is compact. done
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: November 19th 2011, 06:32 AM
  2. Finite union of compact sets is compact
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 8th 2011, 07:43 PM
  3. the intersection of a collection of compact sets is compact
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 28th 2010, 01:58 PM
  4. uniformly on compact subsets
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 15th 2009, 01:44 PM
  5. compact subsets
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 11th 2007, 05:21 PM

Search Tags


/mathhelpforum @mathhelpforum