Hello, there is a statement as following:


If every point of X is a G\_delta and X is T\_1, then take Y = set of X, 
plus the topology generated by all open sets needed to prove G\_delta-ness of every singleton, 
plus the cofinite topology, then Y is a condensation of X (using identity) and is first countable 
by construction.


So a T\_1 space X has a first countable T\_1 condensation iff every point is a G-delta


I have some questions about this statement which need to be solved.


1, why the space Y is first countable?


2, if we delete the cofinite topology, the space Y is still the first countable T\_1 condensation?


3, how to make the topology generated on Y be first countable and hausdorff, if X is hausdorff and 


every point of X is a G\_delta.



Topology generated by the collection of open sets
So we start with $(X, \mathcal{T})$, a $T\_1$ space in which every point is a $G\_{\delta}$, as witnessed by open sets $U\_n(x)$, $n \in \mathbb{N}$, $x \in X$. W.l.o.g. we can take these sets to be decreasing.


The topology generated by all these sets we call $\mathcal{T}'$, say, and it is $T\_1$, because for every $ x \neq y $, there is some $U\_n(x)$ that does not contain $y$ (or else $y$ would be in their intersection, for all $n$, and this intersection is precisely $\{x\}$), and this witnesses the $T\_1$ property ($\{y\}$ is closed, by this argument).


A base for this topology is given by all finite intersections of the sets $U\_n(x)$. It's not clear how this topology would be first countable; the obvious candidate for a local base at $x$ is all sets $U\_n(x)$, but if $z$ is in $U\_n(x) \cap U\_m(y)$, I don't see how we automatically get some $U\_m(z)$ inside it. So in general, this might not work as advertised. 


This might be remedied by using elementary submodels (see e.g. <http://www.math.uncc.edu/~adow/Ftp/Intro.Elem.Subm/elem.ps>) I think. This would also then do the Hausdorff case, which this construction (even if it could be made to work) does not always give, e.g. if we take a countable nowhere first countable space and use the cofinite topology on it. This is a $T\_1$ condensation obtained by the described construction, which is not Hausdorff.


