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Let $X$ be a normal Hausdorff space. Let $A_1$, $A_2$, and $A_3$ be closed subsets of $X$ which are pairwise disjoint. Then there always exists a continuous real valued function $f$ on $X$ such that $f(x) = a_i$ if $x$ belongs to $A_i$, $i=1,2,3$

  1. iff each $a_i$ is either 0 or 1.

  2. iff at least two of the numbers $a_1$, $a_2$, $a_3$ are equal.

  3. for all real values of $a_1$, $a_2$, $a_3$.

  4. only if one among the sets $A_1$, $A_2$, $A_3$ is empty.

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please help how to tackle these types of problems.and also suggest a good topology problem book.thanks –  poton Aug 12 '12 at 2:23
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You may consider accepting some answers on other questions that you've asked. This might motivate more people to try to answer. –  Andrew Aug 12 '12 at 2:26
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how to accept please help. –  poton Aug 12 '12 at 2:35
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@poton Go to a question which you previously asked and scroll down to an answer which you find to be correct/helpful. There should be a checkmark to the left of the question which you can click on. After clicking it, the check mark should become green, indicating your acceptance of the answer. meta.math.stackexchange.com/questions/3286/… –  Andrew Aug 12 '12 at 2:40
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@poton I suggest you read the faq. –  Alex Becker Aug 12 '12 at 3:13
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1 Answer

up vote 1 down vote accepted

The third choice is correct. Recall Urysohn's lemma, which states that a space is normal iff disjoint closed sets can be separated by a function. Since $A_1$ and $A_2$ are disjoint closed sets, we have a continuous function $g:X\to [0,1]$ such that $g(x)=0$ for $x\in A_1$ and $g(x)=1$ for $x\in A_2$. Since $A_2\cup A_3$ and $A_3$ are closed disjoint sets, we have a continuous function $h:X\to [0,1]$ such that $h(x)=0$ if $x\in A_1\cup A_2$ and $g(x)=1$ if $x\in A_3$. Thus we have the continuous function $(g+2h):X\to \mathbb R$ which satisfies $$(g+2h)(x)=\begin{cases} 0 &\text{if } x\in A_1\\ 1 &\text{if } x\in A_2\\ 2 &\text{if } x\in A_3\\ \end{cases}$$ and so composing this with your favorite function $p:\mathbb R\to\mathbb R$ which sends $0$ to $a_1$, $1$ to $a_2$, and $2$ to $a_3$ gives the desired function $f$.

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