pi-base
Variations on the Frechet-Urysohn property
There are a few subtly different strengthenings of the Frechet-Urysohn property (P80) that would be interesting to have in pi-base. In particular:
- strictly Frechet (Gerlits & Nagy 1982: https://doi.org/10.1016/0166-8641(82)90065-7)
- strongly Frechet (Siwiec 1971: https://doi.org/10.1016/0016-660X(71)90120-6)
The implications are: first countable ==> strictly Frechet ==> strongly Frechet ==> Frechet-Urysohn.
Other references:
- https://mathoverflow.net/questions/264857
- https://math.stackexchange.com/questions/756159
- [G] Gruenhage: http://webhome.auburn.edu/~gruengf/preprints/frechprodd.pdf
- [ABD] Aurichi, Bella, Dias 2016: https://arxiv.org/abs/1603.09715
It says on page 2 of [G] that strongly Frechet is the same as $\alpha_4$-FU (= $\alpha_4$ + Frechet-Urysohn). Mentions various connections with $\alpha_i$ properties, and various other names. See also paragraph preceding Theorem 2.17 in [ABD].
Some example spaces to illustrate:
- Frechet, not strongly Frechet: quotient space of $\mathbb R$ with $\mathbb N$ identified to a point
- Frechet, not strongly Frechet: countable sequential fan $S_\omega$
- strongly Frechet, not strictly Frechet: [ABD] Example 2.10 and Fact 2.16.
- strictly Frechet, not first countable: ?
It seems that even the often mentioned first two are not in pi-base.
Just dropping a related comment here. It appears that the definition of "strong Fréchet-Urysohn" appearing in https://en.wikipedia.org/wiki/Fr%C3%A9chet%E2%80%93Urysohn_space is actually "strictly" since it doesn't specify that the sequence of the A_n doesn't have to be descending.
Good observation. That does not surprise me. That section in wikipedia was added by editor Mgkrupa, without any source whatsoever. That has to be taken with a grain of salt, and someone will have to eventually fix this in wikipedia.