1. ## Compact Sets

I am confused about how to approach the following problem:

Assume$\displaystyle S \subset R^n$ is a compact set, and $\displaystyle V \subset R^m$ is compact.
Prove that $\displaystyle S \times V$ is compact in $\displaystyle R^{n+m}.$

First of all, what does it mean by S * V? I do not understand this notation. I have seen this notation used for defining regions like [a, b] $\displaystyle \times$ [c, d] defines region for integration or something. But not in the context of sets. How would I attempt this problem?

2. Originally Posted by Rozaline
I am confused about how to approach the following problem:

Assume$\displaystyle S \subset R^n$ is a compact set, and $\displaystyle V \subset R^m$ is compact.
Prove that $\displaystyle S \times V$ is compact in $\displaystyle R^{n+m}.$

First of all, what does it mean by S * V? I do not understand this notation. I have seen this notation used for defining regions like [a, b] $\displaystyle \times$ [c, d] defines region for integration or something. But not in the context of sets. How would I attempt this problem?
First, it is NOT "S*V", it is "$\displaystyle S\times V$" as you give in the problem itself. I don't know of any meaning for "S*V" for sets but $\displaystyle S\times V$ is the "Cartesian product", the set of ordered pairs, (a, b), such that the first member, a, is from S and the the second member, b, is from V.

([a, b] and [c, d] are sets. They are the sets, in the number line, $\displaystyle \{x| a\le x\le b\}$ and $\displaystyle \{y|c\le y\le d\}$. and $\displaystyle [a, b]\times [c, d]$ is the set in the plane $\displaystyle \{(x, y)|a\le x\le b, c\le y\le d\}$.)

And you will need information about the "product topology" to answer this question! A set of points, {(x,y)}, in $\displaystyle A\times B$ is open if and only if the set of all "x" in the pairs is open in A and the set of "y" in the pairs is open in B.

3. Originally Posted by Rozaline
I am confused about how to approach the following problem:

Assume$\displaystyle S \subset R^n$ is a compact set, and $\displaystyle V \subset R^m$ is compact.
Prove that $\displaystyle S \times V$ is compact in $\displaystyle R^{n+m}.$

First of all, what does it mean by S * V? I do not understand this notation. I have seen this notation used for defining regions like [a, b] $\displaystyle \times$ [c, d] defines region for integration or something. But not in the context of sets. How would I attempt this problem?

You don't know what cartesian product of sets S x V is?? but you must master this before you attempt to study topology, otherwise it is gonna be a very long course for you.

S x V = { (s,v) ; s in S, v in V }

Tonio