A countable dense family of continuous functions on a compact metric space

By [F1] fix a countable dense subset $D$ of $K$. It is nonempty: otherwise no ball around a point of the nonempty space $K$ would meet it. By [F2] enumerate it as $(x_j{)}_{j\ge 1}$. Put $h_j(x)=d(x,x_j)$. The triangle inequality gives $\mathrm{\mid }{h}_j(x)-h_j(y)\mathrm{\mid }\le d(x,y)$, so each $h_j$ is continuous. Every continuous real function $v$ on $K$ is bounded: the open sets $\{\mathrm{\mid }v\mathrm{\mid }<m\}$ for positive integers $m$ cover $K$, and a finite subcover yields a bound. Thus all supremum norms below are finite.

Let $A$ be the real algebra of finite polynomials in the $h_j$ and the constant function $1$, and let $Q$ be its subset with rational coefficients. These are continuous functions, since finite sums and products of continuous real functions are continuous. The set $Q$ is nonempty and at most countable: monomials are coded by finite lists of natural indices (the empty list codes $1$), and a polynomial is coded by a finite list of pairs consisting of a rational coefficient and such a monomial. Induction using [F4], and then [F5] over list lengths, makes both coding sets countable; their evaluation images are countable by composing an enumeration and using [F2]. This uses [F3] for the rational entries. The algebra $A$ separates distinct $x,y$: choose $x_j$ with $d(x,x_j)<d(x,y)\mathrm{/}3$; then $h_j(y)>2d(x,y)\mathrm{/}3>h_j(x)$.

Write $H=\bar{A}$ in the supremum metric. It is a real vector space: if $u,v\in H$, choose approximants $a,b\in A$ within any prescribed positive errors and use $\mathrm{\parallel }(u+v)-(a+b){\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }u-a{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }v-b{\mathrm{\parallel }}_{\mathrm{\infty }}$, with the analogous scalar estimate. It is closed by the definition of closure. By [F6] it is closed under finite maximum and minimum. In applying that lemma, countable choice supplies its sequences of algebra approximants and polynomial approximants; the later diagonal indices can be taken least eligible integers. No arbitrary family indexed by $K$ is selected.

Fix $f\in C(K,\mathbb{R})$ and $\eta >0$. For a fixed $x\in K$, use the entire set $A_x=\{u\in A:u(x)=f(x)\}$. The open sets $\{u>f-\eta \}$, for all $u\in A_x$, cover $K$: at $y\mathrm{\ne }x$, [F7] supplies one interpolant taking the values $f(x),f(y)$, while the constant $f(x)$ handles $y=x$. By [F8] finitely many of these open sets cover $K$; choose a representing function for each of these finitely many sets and form their maximum $g$. Then $g\in H$, $g(x)=f(x)$, and $g>f-\eta$ throughout $K$. Only finitely many existential witnesses were needed for this fixed $x$.

Use the entire set $G=\{g\in H:g>f-\eta \text{ on }K,g(x)=f(x)\text{ for some }x\in K\}$. The preceding step proves that the open sets $\{g<f+\eta \}$, for all $g\in G$, cover $K$, without selecting one $g$ for each $x$. A finite subcover and finitely many representatives give $g_1,\dots ,g_s$; their minimum $v\in H$ satisfies $f-\eta <v<f+\eta$ pointwise, hence $\mathrm{\parallel }v-f{\mathrm{\parallel }}_{\mathrm{\infty }}\le \eta$. Given any $\delta >0$, take $\eta =\delta \mathrm{/}3$ and an $a\in A$ with $\mathrm{\parallel }a-v{\mathrm{\parallel }}_{\mathrm{\infty }}<\delta \mathrm{/}3$. Then $\mathrm{\parallel }a-f{\mathrm{\parallel }}_{\mathrm{\infty }}<\delta$. This proves density of $A$, including when $K$ is a singleton, for which constants alone interpolate.

For $a={\sum }_{i=1}^r{c}_i{m}_i\in A$, where the $m_i$ are monomials, let $M_i=\mathrm{\parallel }{m}_i{\mathrm{\parallel }}_{\mathrm{\infty }}$. If $r=0$, then $a=0\in Q$. Otherwise, given $\delta >0$, use [F9] to choose rational $q_i$ with $\mathrm{\mid }{q}_i-c_i\mathrm{\mid }<\delta \mathrm{/}[r(1+M_i)]$. The finite sum $q={\sum }_i{q}_i{m}_i$ belongs to $Q$ and satisfies $\mathrm{\parallel }a-q{\mathrm{\parallel }}_{\mathrm{\infty }}\le {\sum }_i\mathrm{\mid }{c}_i-q_i\mathrm{\mid }{M}_i<\delta$, even when some $M_i=0$. Approximate $f$ by $a$ within $\epsilon \mathrm{/}2$ using the preceding step, and $a$ by $q$ within $\epsilon \mathrm{/}2$. An enumeration of the nonempty countable $Q$ by [F2] is the required $(f_j)$. Countable choice has been used for the dense subset, the countable-union theorem, and the sequences in [F6]; all cover selections in the local density proof were finite.