Subspace of Product Space is Homeomorphic to Factor Space/Product with Singleton/Proof 1

Theorem

Let $T_1$ and $T_2$ be non-empty topological spaces.

Let $b \in T_2$.

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

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

Then:

$T_1$ is homeomorphic to the subspace $T_1 \times \set b$ of $T_1 \times T_2$

$T_1$ is homeomorphic to the subspace $\set b \times T_1$ of $T_2 \times T_1$

Proof

From Finite Cartesian Product of Non-Empty Sets is Non-Empty both $T_1 \times T_2$ and $T_2 \times T_1$ are both non-empty.

The conclusions follow immediately from Subspace of Product Space is Homeomorphic to Factor Space.

$\blacksquare$