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.