A space that fails to be first countable at every point

It is quite easy to derive a space that failes to have a countable base at one point. We can take a metric space and collapse a set to one point. The resulting quotient space may fail to be first countable at the identified point but is still first countable at every other point. See here and here for such an example. This is a famous quotient space called the sequential fan. Another famous example of a quotient space is the Arens’ space which fails to be first countable at only one point (see..