An example of a non first countable Fréchet-Urysohn space?

Question

As the head title says, I need a Fréchet-Urysohn space but not first countable, (on the way, a good Text book to follow). Thanks.

Answer 1

EDIT: Question was changed to specify "not first countable."

An example is apparently constructed here: http://dml.cz/dmlcz/140083

(I didn't actually read this paper.)

Answer 2

I am posting as a CW answer some things that have been mentioned it the comment. Feel free to edit it further.

Such example is given in Arthur Fischer's answer here.
The example mentioned by Arthur Fischer is also mentioned in Engelking's General topology in Examples 1.4.17 and 1.6.18. This book has a chapter devoted to Fréchet-Urysohn and sequential spaces, which is fairly detailed.

Dan Ma has on his blog a series of posts on sequential spaces, starting here. He includes examples showing that none of the implications fist countable $\Rightarrow$ Fréchet $\Rightarrow$ sequential $\Rightarrow$ k-space cannot be reversed.

Answer 3

Another example which has not been mentioned so far:

Let $(X,\tau)$ denote the real line with the cofinite topology. The uncountability prevents $X$ from being first-countable.

What about convergence? We can distinguish three cases:

• If $(x_n)_n$ assumes no value infinitely often, then it converges to every point in $X$.
• If $(x_n)_n$ assumes exactly one value infinitely often, then it converges to this point and no other point.
• If $(x_n)_n$ assumes several values infinitely often, then it diverges.

The result is an immediate consequence of
Lemma: Let $x\in X$ and $(x_n)_n$ a sequence in $X$. Then $(x_n)_n$ converges to $x$ if and only if for every $y\ne x$ we have $x_n=y$ for only finitely many $n\in\Bbb N$.

It is now easy to prove that $(X,\tau)$ is Fréchet-Urysohn.