the problem says:
Show that a continuous open image of completely regular space need not be completely regular.
I do not know how to start, i can not think of anything outstanding, any suggestions
2026-03-28 01:25:49.1774661149
A continuous open image of completely regular space need not be completely regular
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If completely regular includes $T_1$, then the standard open map counterexample already works: let $f: \mathbb{R} \to \{0,1\}$, where $\{0,1\}$ has the indiscrete/trivial topology, given by $f(x) = 0$ for $x$ rational and $f(x) =1$ otherwise. Then $f$ is continuous and open, but $\{0,1\}$ is not $T_1$ or any higher $T_i$-axiom.
Another (non-continuous) example is the identity map from $\mathbb{R}$ to $\mathbb{R}_K$ (the $K$-topology from Munkres, which is the smallest topology on the reals that is finer than the usual topology and which makes $K = \{\frac{1}{n}: n \in \mathbb{N}\}$ closed, and which is not regular or completely regular).