I am writing an expository work. And I need classical references (books or articles) that simultaneously proof the three classical results below. Any suggestion?
Theorem. Let $(\mathbb{X},d_\mathbb{X})$ and $(\mathbb{Y},d_\mathbb{Y})$ be two metric spaces, and let $f:\mathbb{X}\to \mathbb{Y}$ be continuous. If $\mathbb{X}$ is compact, then $f$ is uniformly continuous.
$\quad$
Theorem. Let $(\mathbb{X},d_{\mathbb{X}})$ be a compact metric space and let $f:\mathbb{X}\to \mathbb{R}$ be continuous. Then $f$ has both a minimum and a maximum on $\mathbb{X}$.
$\quad$ Let $\mathbb{X},\mathbb{Y}$ be two topological spaces. For a subset $D$ of $\mathbb{X}$ we denote by $\mathop{\mathrm{int}}(D)$, $\partial D$ and $\mathop{\mathrm{ext}}(D)=X\backslash (D\cup \partial D)=\mathop{\mathrm{int}}(X\backslash D)$ its interior, boundary and exterior respectively. A function $f:\mathbb{X}\to \mathbb{Y}$ is said to be continuous with respect to $D\subseteq \mathbb{Y}$ if $f^{-1}(D)$ is an open set in $\mathbb{X}$. $\quad$
Theorem. Let $\mathbb{X}$ be a connected topological space and let $\mathbb{Y}$ be a topological space. Let $D\subseteq \mathbb{Y}$. If $f:X\to Y$ is continuous with respect to both $\mathop{\mathrm{int}}(D)$ and $\mathop{\mathrm{ext}}(D)$, and if there are $a,b\in \mathbb{X}$ such that $f(a)\in D$ and $f(b)\notin D$, then there exists $x\in\mathbb{ X}$ such that $f(x)\in \partial D$. In particular this is true if $f$ is continuous.
When $\mathbb{R} = \mathbb{Y} $ this result is sometimes known customs theorem.