What does $Bbb S^{n-1}times mathbb{R}$ stand for?

Let $ngeqslant 1,f: Bbb R^n – { 0 } to S^{n – 1} times Bbb R,xto(frac{x}{||x||},ln(||x||))$ is a homeomorphism which the inverse is $f^{-1}:S^{n-1}timesmathbb{R}tomathbb{R}^n-{0},(y,t)to e^{t}y$.

I was reading this example but I cannot understand what $S^{n-1}timesmathbb{R}$ means. I suspect that $S^{n-1}$ is the counter dominion of the function. y

Questions:

1) Is $S^{n-1}$ the counter-dominion of f?

2) Why is the counter dominion $S^{n-1}$? Where does the $n-1$ comes from?

Thanks in advance!

Answer

Answer $Bbb S^{n-1}$ is the unit sphere in $Bbb R^n$ ie
$$Bbb S^{n-1} ={xiinBbb R^n: |xi|=1}$$
$n-1$ represent the DIMENSION OF $Bbb S^{n-1}$ as a MANIFOLD
precisely,
$$dim Bbb S^{n-1}= n-1, ~~~~~ dim Bbb R^{n} =n$$

For instance, in dimension 2, i.e in $Bbb R^{2}$
the unit circle is defined
$$Bbb S^{1} ={xiinBbb R^2: |xi|=1} equiv {e^{itheta}: thetain[0,2pi)}$$
is of dimension one. $dimBbb S^1 =1$ roughly speaking you see $Bbb S^1$ as the real line $Bbb R$.

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