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I need a space with countable tightness which is not a Fréchet space. In this space, I am searching for a point with no deleted neighborhood consisting entirely of P-points.

(A P-point is a point $x \in X$ such that for every $G_\delta$ set $O$ containing $x$, $x \in \operatorname{int}(O)$ or equivalently $M_x = O_x$, i.e every fixed prime z-filter that contains $x$ is a z-ultrafilter.)

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Can you tell us that your example is not listed in Steen & Seebach? See austinmohr.com/home/?page_id=146 –  Gerald Edgar Aug 21 '13 at 12:28
    
Vahideh, please consult Chapters 15 and 16 of J. Kąkol, W. Kubiś and M. López-Pellicer, Descriptive Topology in Selected Topics of Functional Analysis, Developments in Mathematics, Vol. 24, Springer Science+Business Media, New York 2011. You will find there a large supply of examples you seek (mostly in the realm of function spaces), and it is very likely that the answer to your question can be deduced from the results gathered in this book. –  Tomek Kania Aug 21 '13 at 13:11
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Why was the definition of a $P$-point given in the question, but not the definition of tightness and a Frechet space? –  Joseph Van Name Aug 21 '13 at 16:19
    
Thanks for your hints. (for Joseph:) I think P-point may be is not well known for Pure topologists. –  Vahideh Bagheri Sep 2 '13 at 7:21

1 Answer 1

up vote 2 down vote accepted

First note that in a space with countable tightness: $P$-point $\Leftrightarrow$ weak $P$-point $\Leftrightarrow$ Isolated point. So you are looking for a point for which every deleted neighborhood contains a non-isolated point. You can do a lot better:

Let $X$ be a countable maximal space (i.e. such that the topology on $X$ is maximal among those topologies which have no isolated points). This is an easy application of Zorn´s lemma, starting with any countable space which has no isolated points. Then:

1) $X$ is countable (in particular it has countable tightness),

2) No point of $X$ is a limit point of two disjoint subsets of $X$ (in particular it has no non-trivial convergent sequences and therefore it is not Fréchet),

3) $X$ has no isolated points (so any point has the property that you want).

Property 2 is not so easy to check, but you can look at "Applications of maximal topologies" by E.K. van Douwen, where he proves (see Theorem 2.2) that for crowded spaces (i.e. spaces with no isolated points) property 2 (which he calls perfectly disconnected) is equivalent to being a maximal space. In the same paper, van Douwen shows how you can construct such a space which is also regular (this is not a simple application of Zorn´s lemma anymore).

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Thank u dear. This example was that I wish. Do you know QP-spaces? (spaces X that C(X) has no prime z-ideal except primes and maximals.) This space is a poinwise QP-space. –  Vahideh Bagheri Sep 2 '13 at 8:07
    
[quote]First note that in a space with countable tightness: $P$-point $\Leftrightarrow$ weak $P$-point $\Leftrightarrow$ Isolated point.[\quote] Dear Ramiro de la Vega, I don't find above proposition in literatures. please refer me to that refrense. –  Vahideh Bagheri Sep 6 '13 at 15:04
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@VahidehBagheri: If $x$ is a weak $P$-point then $x$ is not in the closure of any countable $A \subseteq X \setminus \lbrace x \rbrace$ and therefore (using countable tightness) it is not in the closure of $X \setminus \lbrace x \rbrace$; that is, $x$ is isolated. The other implications: Isolated $\Rightarrow$ $P$-point $\Rightarrow$ weak $P$-point are obviously true in any space (never mind the tightness). –  Ramiro de la Vega Sep 6 '13 at 15:37
    
Thank you Dear Vega –  Vahideh Bagheri Sep 6 '13 at 20:01

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