S226: Gluing of Fortissimo and Sierpinski spaces#1810
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I think it may be slightly easier to avoid quotients all together (or at least as one possible construction)? I.e. a space of size aleph1 with two special points p and q such that every points except p is isloated and ....
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New S226: Fortissimo space of size$\aleph_1$ and Sierpinski space glued at their limit points.
This provides an example of both of these:
π-Base, Search for
Well-based + Has countable $\pi$-character + ~First countableπ-Base, Search for
Almost discrete + ~ Sequentially discrete + ~ Countably tightAs discussed in #1712 (comment) and #1807 (comment).