Results 1 to 5 of 5

Thread: Normal space and Tietze/Urysohn

  1. #1
    CSM
    CSM is offline
    Junior Member
    Joined
    Oct 2010
    Posts
    56

    Lightbulb Normal space and Tietze/Urysohn

    I’m having a really hard time with this one. On a conceptual level and on a technical level. I’ve got my three topology books here, but I can’t seem to make anything out of it. Can somebody enlighten me?

    Let $\displaystyle T$ be a topological space. And let $\displaystyle T$ be normal and $\displaystyle \{C_1, C_2,…,\}$ a countable family of closed subsets of $\displaystyle T$. Suppose that every point of $\displaystyle T$ has an neighbourhood $\displaystyle U$ such that $\displaystyle U\cap C_i\not=\emptyset$ for at most one $\displaystyle C_i$. (Note in perticular that $\displaystyle C_i\cap C_j=\emptyset$ when $\displaystyle i\not= j$.)

    Prove: there are open sets $\displaystyle U_1,U_2,… $ such that $\displaystyle C_i\subset U_i$ for each $\displaystyle i$ and such that $\displaystyle \overline{U_i}\cap\overline{U_j}=\emptyset$ when $\displaystyle i\not= j$
    (Use Tietze Extension Theorem or Urysohn’s lemma)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    Quote Originally Posted by CSM View Post
    Iím having a really hard time with this one. On a conceptual level and on a technical level. Iíve got my three topology books here, but I canít seem to make anything out of it. Can somebody enlighten me?

    Let $\displaystyle T$ be a topological space. And let $\displaystyle T$ be normal and $\displaystyle \{C_1, C_2,Ö,\}$ a countable family of closed subsets of $\displaystyle T$. Suppose that every point of $\displaystyle T$ has an neighbourhood $\displaystyle U$ such that $\displaystyle U\cap C_i\not=\emptyset$ for at most one $\displaystyle C_i$. (Note in perticular that $\displaystyle C_i\cap C_j=\emptyset$ when $\displaystyle i\not= j$.)

    Prove: there are open sets $\displaystyle U_1,U_2,Ö $ such that $\displaystyle C_i\subset U_i$ for each $\displaystyle i$ and such that $\displaystyle \overline{U_i}\cap\overline{U_j}=\emptyset$ when $\displaystyle i\not= j$
    (Use Tietze Extension Theorem or Urysohnís lemma)
    Each $\displaystyle x\in C_1$ has an open neighbourhood $\displaystyle U_x$ with $\displaystyle U_x\cap C_n = \varnothing$ for all n>1. Let $\displaystyle V_1 = \textstyle\bigcup_{x\in C_1} U_x$, and use Urysohn or Tietze to get an open set $\displaystyle U_1$ with $\displaystyle C_1\subseteq U_1\subseteq\overline{U_1}\subseteq V_1.$

    Now use induction. Suppose that open sets $\displaystyle U_1,\ldots,U_n$ with disjoint closures have been chosen so that $\displaystyle C_k\subseteq U_k$ and $\displaystyle \overline{U_k}\cap C_r = \varnothing$ for $\displaystyle 1\leqslant k\leqslant n$ and for all $\displaystyle r>n.$ Show that each $\displaystyle x\in C_{n+1}$ has an open neighbourhood $\displaystyle U_x$ disjoint from $\displaystyle C_r$ for all r>n, and also disjoint from $\displaystyle \overline{U_k} (1\leqslant k\leqslant n).$ As for the case n=1, define open sets

    $\displaystyle C_{n+1}\subseteq U_{n+1}\subseteq\overline{U_{n+1}}\subseteq V_{n+1} = \bigcup_{x\in C_{n+1}} U_x,$

    and check that $\displaystyle U_{n+1}$ has the right properties to continue the induction.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    CSM
    CSM is offline
    Junior Member
    Joined
    Oct 2010
    Posts
    56
    Quote Originally Posted by Opalg View Post
    Each $\displaystyle x\in C_1$ has an open neighbourhood $\displaystyle U_x$ with $\displaystyle U_x\cap C_n = \varnothing$ for all n>1. Let $\displaystyle V_1 = \textstyle\bigcup_{x\in C_1} U_x$, and use Urysohn or Tietze to get an open set $\displaystyle U_1$ with $\displaystyle C_1\subseteq U_1\subseteq\overline{U_1}\subseteq V_1.$

    Now use induction. Suppose that open sets $\displaystyle U_1,\ldots,U_n$ with disjoint closures have been chosen so that $\displaystyle C_k\subseteq U_k$ and $\displaystyle \overline{U_k}\cap C_r = \varnothing$ for $\displaystyle 1\leqslant k\leqslant n$ and for all $\displaystyle r>n.$ Show that each $\displaystyle x\in C_{n+1}$ has an open neighbourhood $\displaystyle U_x$ disjoint from $\displaystyle C_r$ for all r>n, and also disjoint from $\displaystyle \overline{U_k} (1\leqslant k\leqslant n).$ As for the case n=1, define open sets

    $\displaystyle C_{n+1}\subseteq U_{n+1}\subseteq\overline{U_{n+1}}\subseteq V_{n+1} = \bigcup_{x\in C_{n+1}} U_x,$

    and check that $\displaystyle U_{n+1}$ has the right properties to continue the induction.
    Thank you very much for explaining me.

    But I can't grasp it totally.
    In you first sentence you tell me to use Urysohn/Tietze to get an open set $\displaystyle U_1$ with $\displaystyle C_1\subseteq U_1\subseteq\overline{U_1}\subseteq V_1.$
    How is Tietze/Urysohn used there?

    BTW. I've got a lemma + proof in my book that says: A topological space $\displaystyle X$ is normal if and only if for each closed subset $\displaystyle E$ of $\displaystyle X$ and each open set $\displaystyle W$ containing $\displaystyle E$ there exists an open set $\displaystyle U$ containing $\displaystyle E$ such that $\displaystyle \overline{U}\subset W$.
    Isn't that just what I need for getting that set $\displaystyle U_1$

    I thought I had to do something with functions and inverse images.... by the hint...?

    I really appreciate the time you take to help me.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    CSM
    CSM is offline
    Junior Member
    Joined
    Oct 2010
    Posts
    56
    I'd really like to understand the way you tried to solve this, nevertheless I'll show another attempt at solving it (helped by others):


    $\displaystyle \bullet$ Let $\displaystyle C=\bigcup_n C_n$.
    Then $\displaystyle C$ is closed. Because if you take a $\displaystyle x$ not in $\displaystyle C$, we'll find a neighbourhood $\displaystyle O_x$ of $\displaystyle x$ intersecting at most one $\displaystyle C_n$, say $\displaystyle C_k$. Now $\displaystyle O_x\backslash C_k$ is open, it contains $\displaystyle x$ (for $\displaystyle x$ is not in $\displaystyle C_k$ because it is not in $\displaystyle C$) and misses all $\displaystyle C_n$ en thus $\displaystyle C$).

    $\displaystyle \bullet$ Each $\displaystyle C_n$ is still closed in $\displaystyle C$, (...), and for defining a continuous function, we only need to define a continuous function $\displaystyle f_n$ on every $\displaystyle C_n$ and combine them ($\displaystyle f$ will be continuous again).

    $\displaystyle \bullet$ Define $\displaystyle f_n$ on $\displaystyle C_n$ as being constant $\displaystyle n$. Extend $\displaystyle f$ to $\displaystyle X$ (Tietze) and use the $\displaystyle f^{-1}(n-\frac{1}{3},n+\frac{1}{3})$, $\displaystyle n\in N$ as the required $\displaystyle U_n$.

    -$\displaystyle f(C_1)=1, f(C_2)=2,...$ etc. So $\displaystyle f^{-1}(\frac{2}{3},1\frac{1}{3})=C_1$ and $\displaystyle f^{-1}(1\frac{2}{3},2\frac{1}{3})=C_2$ so it easy to show that their closures are disjoint.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    That looks like a very good way to do the question.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof of Tietze Extension Theorem in Munkres
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Sep 13th 2011, 06:52 AM
  2. Tietze Extension Theorem
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Aug 28th 2011, 04:49 AM
  3. A normal Moore space is completely normal
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Mar 19th 2011, 12:52 PM
  4. Normal space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Dec 28th 2009, 05:18 PM
  5. On the Tietze extension theorem
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: Sep 2nd 2009, 09:03 AM

Search Tags


/mathhelpforum @mathhelpforum