# Empty Mapping is Unique

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## 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