Subspace of Product Space is Homeomorphic to Factor Space/Product with Singleton
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 1
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.
$\blacksquare$
Proof 2
The conclusions are symmetrical.
Without loss of generality, therefore, it will be shown that $T_1$ is homeomorphic to the subspace $T_1 \times \set b$ of $T_1 \times T_2$.
Let $f: T_1 \to T_1 \times \set b$ be defined as:
- $\map f x = \tuple {x, b}$
Lemma
- $f$ is a bijection.
$\Box$
$f^{-1}$ is a restriction to the subspace $T_1 \times \set b$ of the projection $\pr_1$ of $T_1 \times T_2$ onto $T_1$.
From Projection from Product Topology is Continuous , $\pr_1$ is a continuous.
From Restriction of Continuous Mapping is Continuous, $f^{-1}$ is a continuous.
It is to be shown that $f$ is continuous.
Let $x \in T_1$.
Let $U$ be an open set in $T_1 \times \set b$ such that:
- $\map f x \in U$
By definition of the subspace topology then for some $U'$ open in $T_1 \times T_2$:
- $U' \cap \paren {T_1 \times \set b} = U$
By the definition of the Natural Basis of Product Topology, there exist open sets $V_1$ and $V_2$ in $T_1$ and $T_2$, respectively, such that:
- $\map f x = \tuple {x, b} \in V_1 \times V_2 \subseteq U'$
Then for any $y \in V_1$:
- $\map f y = \tuple {y, b} \in V_1 \times V_2 \subseteq U'$
But:
- $\tuple {y, b} \in T_1 \times \set b$
so:
- $\tuple {y, b} \in U$
Thus if $y \in V_1$, it follows that $\map f y \in U$.
Then $f$ is continuous by definition.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: Exercise $3.9: 19$