Separation Properties Preserved under Topological Product
Theorem
Let $\mathbb S = \family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod_{i \mathop \in I} \struct{S_i, \tau_i}$ be the product space of $\mathbb S$.
Then $T$ has one of the following properties if and only if each of $\struct {S_i, \tau_i}$ has the same property:
If $T = \struct {S, \tau}$ has one of the following properties then each of $\struct {S_i, \tau_i}$ has the same property:
but the converse does not necessarily hold.
Corollary
$T = \struct {S, \tau}$ has one of the following properties if and only if each of $\struct {S_i, \tau_i}$ has the same property:
If $T = \struct {S, \tau}$ has one of the following properties then each of $\struct {S_i, \tau_i}$ has the same property:
but the converse does not necessarily hold.
Proof
$\blacksquare$
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