Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum

Theorem
Let $\BB$ be a set of subsets of $\R$.

Let:
 * $\size \BB < \mathfrak c$

where
 * $\size \BB$ denotes the cardinality of $\BB$
 * $\mathfrak c = \size \R$ denotes continuum.

Let
 * $X = \leftset {x \in \R: \exists U \in \set {\bigcup \GG: \GG \subseteq \BB}: x}$ is local minimum in $\rightset U$

Then:
 * $\size X < \mathfrak c$

Proof
We will prove that:
 * $(1): \quad \size \BB \aleph_0 < \mathfrak c$

where $\aleph_0 = \size \N$ by Aleph Zero equals Cardinality of Naturals.

In the case when $\size \BB = \mathbf 0$ we have by Zero of Cardinal Product is Zero:
 * $\size \BB \aleph_0 = \mathbf 0 < \mathfrak c$

In the case when $\mathbf 0 < \size \BB < \aleph_0$:

In the case when $\size \BB \ge \aleph_0$ we have:

Define:
 * $Y = \leftset {x \in \R: \exists U \in \BB: x}$ is local minimum in $\rightset U$

We will show that $X \subseteq Y$ by definition of subset.

Let $x \in X$.

By definition of $X$:
 * $\exists U \in \leftset {\bigcup \GG: \GG \subseteq \BB}: x$ is local minimum in $\rightset U$


 * $\exists \GG \subseteq \BB: U = \bigcup \GG$

By definition of local minimum:
 * $x \in U$

By definition of union:
 * $\exists V \in \GG: x \in V$

By definition of subset
 * $V \in \BB$

By definition of local minimum
 * $\exists y \in \R: y < x \land \openint y x \cap U = \O$

By Set is Subset of Union:
 * $V \subseteq U$

Then:
 * $\exists y \in \R: y < x \land \openint y x \cap V = \O$

By definition:
 * $x$ is local minimum in $V$

Thus by definition of $Y$
 * $x \in Y$

So
 * $(2): \quad X \subseteq Y$

Define $\family {Z_A}_{A \mathop \in \BB}$ as:
 * $Z_A = \leftset {x \in \R: x}$ is local minimum in $\rightset A$

We will prove that:
 * $(3): \quad Y \subseteq \ds \bigcup_{A \mathop \in \BB} Z_A$

Let $x \in Y$.

By definition of $Y$:
 * $\exists U \in \BB: x$ is local minimum in $U$

By definition of $Z_U$:
 * $x \in Z_U$

Thus by definition of union:
 * $x \in \ds \bigcup_{A \mathop \in \BB} Z_A$

This ends the proof of inclusion.

By Set of Local Minimum is Countable:
 * $\forall A \in \BB: Z_A$ is countable

By Countable iff Cardinality not greater Aleph Zero:
 * $\forall A \in \BB: \size {Z_A} \le \aleph_0$

By Cardinality of Union not greater than Product:
 * $(4): \quad \ds \size {\bigcup_{A \mathop \in \BB} Z_A} \le \size \BB \aleph_0$

Thus: