Existence of non-metrizable CT3 and CT4 spaces

Construct a non-metrizable CT3-space or, more strongly, a non-metrizable CT4-space.

Background

The paper establishes several CT3 and CT4 examples, but all examples presented as CT3 or CT4 in the paper are metrizable. This motivates the unresolved existence question for non-metrizable spaces satisfying these continuous separation axioms.

References

Many questions are put. For example, does there exist a non-metrizable CT3 or CT4 space?

— Axioms of Continuous Separation  (2608.13086 - Yang et al., 13 Aug 2026) in Abstract; Section 6, Problem labelled metr-CT3

We do not know whether or not there exist other metric spaces satisfying the requirements in Theorem \ref{gener-T3}.

— Axioms of Continuous Separation  (2608.13086 - Yang et al., 13 Aug 2026) in Remark following Example Euclidean-space

In particular, we guess that the following problem has an affirmative answer: For a space $X$ with a unique non-isolated point, whether or not is $X$ metrizable if it is $CT_3$?

— Axioms of Continuous Separation  (2608.13086 - Yang et al., 13 Aug 2026) in Section 6, Problem labelled metr-CT3