# Set of Local Minimum is Countable

## Theorem

Let $X$ be a subset of $\R$.

The set:

$\leftset {x \in X: x}$ is local minimum in $\rightset X$

is countable.

## Proof

Define:

$Y := \leftset {x \in X: x}$ is local minimum in $\rightset X$

By definition of $Y$ and definition of local minimum in set:

$\forall x \in Y: \exists y \in \R: y < x \land \openint y x \cap X = \O$

By the Axiom of Choice, define a mapping $f: Y \to \powerset \R$ as:

$\forall x \in Y: \exists y \in \R: \map f x = \openint y x \land y < x \land \map f x \cap X = \O$

We will prove that $f$ is an injection by definition:

Let $x_1, x_2 \in Y$ such that

$\map f {x_1} = \map f {x_2}$

By definition of $f$:

$\exists y_1 \in \R: \map f {x_1} = \openint {y_1} {x_1} \land y_1 < x_1 \land \map f {x_1} \cap X = \O$

and:

$\exists y_2 \in \R: \map f {x_2} = \openint {y_2} {x_2} \land y_2 < x_2 \land \map f {x_2} \cap X = \O$

Then:

$\openint {y_1} {x_1} = \openint {y_2} {x_2}$

Thus $x_1 = x_2$.

This ends the proof of injection.

$(1): \quad \card Y = \card {\map {f^\to} Y} = \card {\Img f}$

where

$\card Y$ denotes the cardinality of $Y$,
$\map {f^\to} Y$ denotes the image of $Y$ under $f$,
$\Img f$ denotes the image of $f$.

We will prove that $\Img f$ is pairwise disjoint by definition.

Let $A, B \in \Img f$ such that

$A \ne B$.

Then by definition of image:

$\exists x_1 \in Y: \map f {x_1} = A$

and

$\exists x_2 \in Y: \map f {x_2} = B$.

By difference of $A$ and $B$:

$x_1 \ne x_2$

By definition of $f$:

$\exists y_1 \in \R: \map f {x_1} = \openint {y_1} {x_1} \land y_1 < x_1 \land \map f {x_1} \cap X = \O$

and:

$\exists y_2 \in \R: \map f {x_2} = \openint {y_2} {x_2} \land y_2 < x_2 \land \map f {x_2} \cap X = \O$

$A \cap B \ne \O$.

$x_1 < x_2$ or $x_1 > x_2$.

In case when $x_1 < x_2$, $x_1 \in \map f {x_2}$ what contradicts with $\map f {x_2} \cap X = \O$.

In case when $x_1 > x_2$, analogically.

This ends the proof that $\Img f$ is pairwise disjoint.

$\Img f$ is countable.

Thus by $(1)$ and Set is Countable if Cardinality equals Cardinality of Countable Set the result:

$Y$ is countable.

$\blacksquare$