Example of Hausdorff and Second Countable Space that is Not Metrizable Does there exist topological space that is Hausdorff and second countable but not metrizable?

Let $X$ be the real line with the topology in which the usual open sets are open, and in addition $U\setminus A$ is open for any $U$ that is open in the usual topology, where $A=\{\frac1n : n=1,2,...\}$. In other words, every point except the origin has its usual neighborhoods, and the basic neighborhoods of the origin are of the form $(-\varepsilon,\varepsilon)\setminus A$. Then $A$ is a closed subspace that cannot be separated by disjoint neighborhoods from the origin, so $X$ is not regular, and not metrizable. $X$ is Hausdorff since its topology is stronger than the usual topology which is Hausdorff. If we take a countable basis for the usual topology together with a countable local  basis at the origin for the new topology, then these two together form a countable basis for the new topology. (This is a standard example, e.g. Ex.1.5.6 in General Topology by R. Engelking.) 
