Topological Subspace of Real Number Line is Lindelöf

Theorem
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $W$ be a non-empty subset of $\R$.

Then $R_W$ is Lindelöf

where $R_W$ denotes the topological subspace of $R$ on $W$.

Proof
Let $\CC$ be a open cover for $W$.

Define $Q := \set {\openint a b: a, b \in \Q}$

Define a mapping $h: Q \to \Q \times \Q$:
 * $\forall \openint a b \in Q: \map h {\openint a b} = \tuple {a, b}$

It is easy to see by definition that
 * $h$ is an injection.

By Injection iff Cardinal Inequality:
 * $\card Q \le \card {\Q \times \Q}$

where $\card Q$ deontes the cardinality of $Q$.

By Rational Numbers are Countably Infinite:
 * $\Q$ is countably infinite.

By definition of countably infinite:
 * there exists a bijection $\Q \to \N$

By definitions of set equality and cardinality:
 * $\card \Q = \card \N$

By Aleph Zero equals Cardinality of Naturals:
 * $\card \Q = \aleph_0$

By Cardinal Product Equal to Maximum:
 * $\card {\Q \times \Q} = \max \set {\aleph_0, \aleph_0} = \aleph_0$

By Countable iff Cardinality not greater than Aleph Zero:
 * $Q$ is countable.

By definition of cover:
 * $W \subseteq \bigcup \CC$

By definition of imion:
 * $\forall x \in W: \exists U \in \CC: x \in U$

By Axiom of Choice define a mapping $f: W \to \CC$:
 * $\forall x \in W: x \in \map f x$

We will prove that
 * $\forall x \in W: \exists A \in Q: x \in A \land A \cap W \subseteq \map f x$

Let $x \in W$.

By definition of open cover:
 * $\map f x$ is open in $R_W$.

By definition of topological subspace:
 * there exists U a subset of $\R$ such that
 * $U$ is open in $R$ and $U \cap W = \map f x$

By definition of $f$:
 * $x \in \map f x$

By definition of open set in metric space:
 * $\exists r > 0: \map {B_r} x \subseteq U$


 * $\openint {x - r} {x + r} \subseteq U$

By Between two Real Numbers exists Rational Number:
 * $\exists q \in \Q: x - r < q < x$ and $\exists p \in \Q: x < p < x + r$

By definition of $Q$:
 * $\openint q p \in Q$

Thus by definition of open real interval:
 * $x \in \openint q p \subseteq \openint {x - r} {x + r}$

By Subset Relation is Transitive:
 * $\openint q p \subseteq U$

Thus by Set Intersection Preserves Subsets/Corollary:
 * $\openint q p \cap W \subseteq \map f x$

By Axiom of Choice define a mapping $f_1: W \to Q$:
 * $\forall x \in W: x \in \map {f_1} x \land \map {f_1} x \cap W \subseteq \map f x$

By definitions of image of set and image of mapping:
 * $\forall A \in \Img {f_1}: \exists x \in W: x \in f_1^{-1} \sqbrk {\set A}$

By Axiom of Choice define a mapping $c: \Img {f_1} \to W$:
 * $\forall A \in \Img {f_1}: \map c A \in f_1^{-1} \sqbrk {\set A}$

Define a mapping $g: \Img {f_1} \to \CC$:
 * $g := f \circ c$

Define $\GG = \Img g$.

Thus $\GG \subseteq \CC$

By definition of image of mapping:
 * $\Img {f_1} \subseteq Q$

By Subset of Countable Set is Countable;
 * $\Img {f_1}$ is countable.

By Surjection iff Cardinal Inequality:
 * $\size {\Img g} \le \size {\Img {f_1} }$

Thus by Countable iff Cardinality not greater than Aleph Zero:
 * $\GG$ is countable.

It remains to prove that
 * $\GG$ is a cover for $W$.

Let $x \in W$.

By definition of $f_1$:
 * $ x \in \map {f_1} x \land \map {f_1} x \cap W \subseteq \map f x$

By definition of image of mapping:
 * $\map {f_1} x \in \Img {f_1}$

Then by definition of $c$:
 * $y := \map c {\map {f_1} x} \in f_1^{-1} \sqbrk {\set {\map {f_1} x} }$

By definition of $f_1$:
 * $y \in \map {f_1} y \land \map {f_1} y \cap W \subseteq \map f y$

By definition of image of set:
 * $\map {f_1} y = \map {f_1} x$

Then by definitions of subset and intersection:
 * $x \in \map f y$

By definition of composition of mappings:
 * $\map f y = \map g {\map {f_1} x}$

By definition of image of mapping:
 * $\map g {\map {f_1} x} \in \GG$

Thus by definition of union:
 * $x \in \bigcup \GG$

Thus the result.