Factor Spaces are T4 if Product Space is T4
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Theorem
Let $\mathbb S = \left\{{\left({S_\alpha, \tau_\alpha}\right)}\right\}$ be a set of topological spaces for $\alpha$ in some indexing set $I$.
Let $\displaystyle T = \left({S, \tau}\right) = \prod \left({S_\alpha, \tau_\alpha}\right)$ be the product space of $\mathbb S$.
Let $T$ be a $T_4$ space.
Then each of $\left({S_\alpha, \tau_\alpha}\right)$ is a $T_4$ space.
Proof
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Sources
- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{I}: \ \S 2$: Functions, Products, and Subspaces
