Examples of Separable Spaces that are not Second-Countable In this post I give an example of a separable space that is not second-countable. What are other good examples?

$\pi$-Base is an online encyclopedia of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It currently lists twenty-one separable spaces that are not second countable. You can learn more about any of these spaces by viewing the search result.
Appert Space
Arens-Fort Space
Deleted Diameter Topology
Deleted Radius Topology
Finite Complement Topology on an Uncountable Space
Half-Disc Topology
Helly Space
Maximal Compact Topology
Niemytzki's Tangent Disc Topology
Novak Space
One Point Compactification of the Rationals
Pointed Rational Extension of the Reals
Rational Extension in the Plane
Rational Sequence Topology
Right Half-Open Interval Topology
Single Ultrafilter Topology
Sorgenfrey's Half-Open Square Topology
Stone-Cech Compactification of the Integers
Strong Parallel Line Topology
Strong Ultrafilter Topology
Uncountable Particular Point Topology
