TITLE: Example of Hausdorff and Second Countable Space that is Not Metrizable
QUESTION [5 upvotes]: Does there exist topological space that is Hausdorff and second countable but not metrizable?

REPLY [3 votes]: $\pi$-Base is a database of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It lists the following fourteen second countable, Hausdorff spaces that are not metrizable. You can learn more about an of them from the search result.
Arens Square
Double Origin Topology
Indiscrete Irrational Extension of $\mathbb{R}$
Indiscrete Rational Extension of $\mathbb{R}$
Irrational Slope Topology
Irregular Lattice Topology
Minimal Hausdorff Topology
Pointed Irrational Extension of $\mathbb{R}$
Prime Integer Topology
Relatively Prime Integer Topology
Roy's Lattice Space
Roy's Lattice Subspace
Simplified Arens Square
Smirnov's Deleted Sequence Topology