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.

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. — Martin Sleziak, Jan 16 '14 at 11:58
You also asked about a good textbook for general topology. You can already find several question on this site about this topic, see the section on topology in List of Generalizations of Common Questions. — Martin Sleziak, Jan 16 '14 at 12:00
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. — Martin Sleziak, Jan 16 '14 at 13:17

score 3 · Answer 1 · answered Jan 16 '14 at 16:19

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.

Of course any uncountable set will do in place of the real line. Your example is not a Hausdorff space but that is easily fixed by making all points but one isolated, e.g., use the topology $\tau={U\subseteq\mathbb R: 0\notin\mathbb R\vee|\mathbb R\setminus U|\lt\aleph_0}$ — bof, Jan 17 '14 at 09:59

score 1 · Answer 2 · edited Apr 13 '17 at 12:20

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.

Daniel McLaury · Answer 3 · 2014-01-16T08:31:02.520

0

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.)

edited Jan 16 '14 at 08:31

answered Jan 16 '14 at 06:49

Daniel McLaury

24,656

Sorry, I edited the main question, I am looking for not first countable Fréchet-Urysohn space. – Jesús Pérez Jan 16 '14 at 07:11
By "anything" did you mean "nothing"? – bof Jan 16 '14 at 07:57
No. For instance a manifold is Frechet-Urysohn. – Daniel McLaury Jan 16 '14 at 08:29

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

3 Answers3