Factor Spaces are T5 if Product Space is T5

Theorem

Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$.

Let $\ds T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$.

Let $T$ be a $T_5$ space.

Then each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_5$ space.

Proof

Let $T$ be a $T_5$ space.

Let $\struct {S_\alpha, \tau_\alpha}$ be arbitrary.

By Subspace of Product Space is Homeomorphic to Factor Space:

$\struct {S_\alpha, \tau_\alpha}$

is homeomorphic to a subspace of $T$.

By $T_5$ Property is Hereditary, this subspace is also $T_5$.

Finally, by $T_5$ Space is Preserved under Homeomorphism:

$\struct {S_\alpha, \tau_\alpha}$

is a $T_5$ space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Functions, Products, and Subspaces