Set Union/Examples/Set of Unbounded Above Open Real Intervals
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Example of Union of Set of Sets
Let $\R$ denote the set of real numbers.
For a given $a \in \R$, let $S_a$ denote the (real) interval:
- $S_a = \openint a \to = \set {x \in \R: x > a}$
Let $\SS$ denote the family of sets indexed by $\R$:
- $\SS := \family {S_a}_{a \mathop \in \R}$
Then:
- $\bigcup \SS = \R$.
Proof
By construction we have that $\SS \subseteq \R$.
It remains to prove that $\R \subseteq \SS$
Aiming for a contradiction, suppose $x \in \R$ such that $x \notin \SS$.
Then by definition of set union:
- $\not \exists a \in \R: x \in S_a$
That is:
- $\forall a \in \R: x \le a$
But then we have:
- $x - 1 \in \R$
leading to:
- $x < x - 1$
which contradicts the properties of real numbers, which leads to:
- $x > x - 1$
Hence the result.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Unions and Intersections