I have a set in $\mathbb{R}^4$ defined as $\mathbf{C} = \{(x, y) \mid x\in\mathbf{P}(y)\subset\mathbb{R}^3, y\in\mathbf{S}\subset\mathbb{R}_+\}$, where $\mathbf{P}(y)$ is a compact set on $\mathbb{R}^3$ for any $y\in\mathbf{S}$, and $\mathbf{S}$ is a compact set on $\mathbb{R}_+$.
For my problem, $\mathbf{P}$ represents a compact geometry in $\mathbb{R}^3$, e.g., polygons and spheres. And $\mathbf{P}(y)$ represents the compact geometry obtained after enlarging $\mathbf{P}$ by a scale of $y$:
$\mathbf{P}(y) = \{yx \mid x\in\mathbf{K}\subset\mathbb{R}^3\}\subset\mathbb{R}^3$.
Where $\mathbf{K}$ is a compact geometry in $\mathbb{R}^3$, e.g., polygons and spheres. And $\mathbf{S}$ is $[10^{-4}, 10^4]$.
How can I show that $\mathbf{C}$ is a compact set on $\mathbb{R}^4$?
Thanks in advance!
Define $\phi : \mathbf{K} \times \mathbf{S} \rightarrow \mathbb{R}^4$ by $\phi(a,b) = (ba, b)$. Then $\phi$ is continous and $\mathbf{C}$ is just the image of the compact set $\mathbf{K} \times \mathbf{S}$ under $\phi$, hence $\mathbf{C}$ is compact.