Subspace of Product Space is Homeomorphic to Factor Space/Product with Singleton/Lemma

Theorem

Let $T_1$ and $T_2$ be non-empty sets.

Let $b \in T_2$.

Let $T_1 \times T_2$ be the Cartesian product of $T_1$ and $T_2$.

Let $f: T_1 \to T_1 \times \set b$ be the mapping defined by:

$\map f x = \tuple {x, b}$

Then:

$f$ is a bijection.

Proof

$f$ is an Injection

Let $x, y \in T_1$.

\(\ds \map f x\)

\(=\)

\(\ds \map f y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \tuple {x, b}\)

\(=\)

\(\ds \tuple {y, b}\)

Definition of $f$

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds y\)

Equality of ordered pairs

Then $f$ is an injection by definition.

$\Box$

$f$ is a Surjection

Let $t \in T_1 \times \set b$

Then:

$\exists x \in T_1: t = \tuple {x, b}$

So:

$\map f x = \tuple {x, b} = t$

Then $f$ is a surjection by definition.

$\Box$

It follows that $f$ is a bijection by definition.

$\blacksquare$