Results 1 to 1 of 1

Thread: Tube lemma generalization

  1. #1
    Oct 2010

    Tube lemma generalization

    So, here's the problem:

    Let A and B be compact subspaces of X and Y, respectively. Let N be an open set in X x Y containing A x B. One needs to show that there exist open sets U in X and V in Y such that A x B $\displaystyle \subseteq$ U x V $\displaystyle \subseteq$ N.

    Here's my try:

    First of all, since N is open, it can be written as a union of basis elements in X x Y, i.e. let N = $\displaystyle \cup U_{i} \times V_{i}$.

    Then we cover A x B with basis elements contained in N, so that $\displaystyle A \times B \subseteq \cup U_{i}' \times V_{i}'$ . Since A and B are compact, so is A x B, and for this cover, we have a finite subcover, so that $\displaystyle A \times B \subseteq \cup_{i=1}^n U_{i}' \times V_{i}'$.

    Now we have the following relation:

    $\displaystyle A \times B \subseteq \cup_{i=1}^n U_{i}' \times V_{i}' \subseteq \cup U_{i} \times V_{i} = N$.

    Now, I'm not sure if this relation holds:

    $\displaystyle \cup_{i=1}^n (U_{i}' \times V_{i}') \cap (\cup U_{i} \times V_{i}) \subseteq \cup_{i=1}^n (U_{i}' \cap (\cup U_{i})) \times \cup_{i=1}^n (V_{i}' \cap (\cup V_{i})) \subseteq N$. If it does, then $\displaystyle U = \cup_{i=1}^n (U_{i}' \cap (\cup U_{i}))$ and $\displaystyle V = \cup_{i=1}^n (V_{i}' \cap (\cup V_{i}))$ are the sets we were looking for.

    If x = (a, b) is in $\displaystyle (\cup_{i=1}^n (U_{i}' \times V_{i}')) \cap (\cup U_{i} \times V_{i})$ then a is in Ui, b is in Vi, for some i, and a is in Ui' and b is in Vi'. So, a is in the intersection of Ui and Ui', for some i, and b is in the intersection of Vi and Vi', for some i, i.e. in their unions, so x is in $\displaystyle (\cup_{i=1}^n (U_{i}' \cap (\cup U_{i}))) \times (\cup_{i=1}^n (V_{i}' \cap (\cup V_{i})))$.

    Does this work?

    Edit: just edited this message, sorry for the math-typing inconvenience before.
    Last edited by Gopnik; Oct 27th 2010 at 11:37 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Thermal expansion of liquid in glass tube
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: Nov 27th 2011, 11:09 PM
  2. Continuity functions (Generalization)
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Apr 28th 2010, 04:22 PM
  3. need help on impedance tube signal analysis
    Posted in the Math Software Forum
    Replies: 0
    Last Post: Feb 28th 2010, 04:03 AM
  4. help on finding the height of the test tube
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Sep 6th 2009, 03:51 AM
  5. Please Need Urgent Help With View Tube Experiment!
    Posted in the Math Topics Forum
    Replies: 3
    Last Post: May 25th 2005, 06:15 PM

Search tags for this page

Click on a term to search for related topics.

Search Tags

/mathhelpforum @mathhelpforum