Space Suggestion
Let $A = \bigcup_{n = 1}^\infty [\frac{1}{3^{n-1}}, \frac{2}{3^n}]$. The Deleted Sequence of Intervals Topology is the topology on $\mathbb{R}$ generated by Euclidean open sets and $A^c$.
Rationale
While this is a new construction (as far as I can tell), it is a natural generation of the Smirnov's deleted sequence topology. Indeed, if each of the intervals in $A$ is replaced by a point in the said interval, then the generated topology would be Smirnov's deleted sequence topology. (This is also the reason I named the topology deleted sequence of intervals topology.) Thus, the two topologies share many properties. However, it is distinct from Smirnov's deleted sequence topology in one important aspect - it is semiregular but not regular. Whence, it provides an example of a submetrizable semiregular space that is not regular. (See Math StackExchange 4996729.)
Relationship to other spaces and properties
This space provides an example satisfying the search https://topology.pi-base.org/spaces?q=Submetrizable+%2B+Semiregular+%2B+not+Regular, which currently has no example on pi-base.
Space Suggestion
Let$A = \bigcup_{n = 1}^\infty [\frac{1}{3^{n-1}}, \frac{2}{3^n}]$ . The Deleted Sequence of Intervals Topology is the topology on $\mathbb{R}$ generated by Euclidean open sets and $A^c$ .
Rationale
While this is a new construction (as far as I can tell), it is a natural generation of the Smirnov's deleted sequence topology. Indeed, if each of the intervals in$A$ is replaced by a point in the said interval, then the generated topology would be Smirnov's deleted sequence topology. (This is also the reason I named the topology deleted sequence of intervals topology.) Thus, the two topologies share many properties. However, it is distinct from Smirnov's deleted sequence topology in one important aspect - it is semiregular but not regular. Whence, it provides an example of a submetrizable semiregular space that is not regular. (See Math StackExchange 4996729.)
Relationship to other spaces and properties
This space provides an example satisfying the search https://topology.pi-base.org/spaces?q=Submetrizable+%2B+Semiregular+%2B+not+Regular, which currently has no example on pi-base.