Empty Mapping is Unique
Jump to navigation
Jump to search
Theorem
For each set $T$ there exists exactly one empty mapping $e: \O \to T$ whose domain is the empty set.
Proof
By definition a mapping from $\O$ to $T$ is a subset of the cartesian product $\O \times T$:
- $e: \O \to T \subseteq \O \times T$
From Empty Mapping is Mapping, we have that the empty mapping from $\O$ to $T$ exists.
From Cartesian Product is Empty iff Factor is Empty it follows that the empty mapping equals the empty set:
- $\O \times T = \O$
The result follows by Empty Set is Unique.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Functions