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If A and B are two closed sets of $R$ is A.B closed? By A.B I mean the set $\sum_{i=1}{^ n} a_ib_i$ where $a_i \in A,b_i\in B,n\in N$ How to view A.B geometrically? I am new to this subject.Sorry if the question sounds something wrong

Learnmore
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  • Are $A$ and $B$ subsets of $\mathbb R$? (or some other set on wich multiplicaton and addition are defined). What exactly is meant by "the set $\sum a_ib_i$"? Make that clear in your question (not in a comment). – drhab Oct 09 '14 at 09:26
  • If we have two sets and $\dim A = a, \dim B = b$, then their product will have $dim (A \times B) = a + b$. So please check again with your definition, as summation you provided is not possible in case $A$ and $B$ have different number of members (if they are finite) or are infinite. – Andrei Rykhalski Oct 09 '14 at 09:34

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Without further clarification from the OP, I am interpreting the question as "$A,B$ are subsets of $\mathbb{R}$, and $A \cdot B =\{ab \,| \, a\in A,b \in B\}$".

If $A,B$ are closed sets in $\mathbb{R}$ with standard topology, then $A\cdot B$ may not be closed in $\mathbb{R}$. An example is $A=\{0\} \cup \{\frac{1}{n} \,|\, n\in \mathbb{N}\}$, and $B=\mathbb{N}$,then $A \cdot B=\{0\} \cup \mathbb{Q}_{+}$, which is not closed in $\mathbb{R}$.

John
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  • Thanks for your interpretation.Can u please help me to view A.B geometrically – Learnmore Oct 09 '14 at 15:08
  • @learningmaths. Note my definition of $A\cdot B$ is different from yours. But in the example I given, they refer to the same set since ${0} \cup \mathbb{Q}+$ is closed under addition. $A \cdot B=\bigcup{b\in B}Ab$, while $Ab$ can be viewed as a scaling of $A$. If $b\neq 0$, $x\to xb$ is homeomorphism, so $Ab$ should be closed. But the problem is that arbitrary union of closed sets may not be closed. – John Oct 09 '14 at 17:39
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First, see user5527's answer.

Now, if $A$ and $B$ are closed and bounded subsets of $\mathbb R$, so that they're compact, then the answer is yes. The Cartesian product $A\times B$ is a compact subset of $\mathbb R^2$, and the set of products $A\cdot B$ is its image under a continuous map.

Chris Culter
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