𝑝-sequentiality and 𝑝-Fréchet-Urysohn property of Franklin compact spaces

Garcia-Ferreira, S.; Malykhin, V.

doi:10.1090/S0002-9939-96-03322-9

$p$-sequentiality and $p$-Fréchet-Urysohn property of Franklin compact spaces
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by S. Garcia-Ferreira and V. I. Malykhin PDF

Proc. Amer. Math. Soc. 124 (1996), 2267-2273 Request permission

Abstract:

Franklin compact spaces defined by maximal almost disjoint families of subsets of $\omega$ are considered from the view of its $p$-sequentiality and $p$-Fréchet-Urysohn-property for ultrafilters $p\in \omega ^*$. Our principal results are the following: CH implies that for every $P$-point $p\in \omega ^*$ there are a Franklin compact $p$-Fréchet-Urysohn space and a Franklin compact space which is not $p$-Fréchet-Urysohn; and, assuming CH, for every Franklin compact space there is a $P$-point $q\in \omega ^*$ such that it is not $q$-Fréchet-Urysohn. Some new problems are raised.

References

A. I. Baškirov, The classification of quotient maps and sequential bicompacta, Dokl. Akad. Nauk SSSR 217 (1974), 745–748 (Russian). MR 0358700
A. I. Baškirov, Maximal almost disjoint systems and Franklin bicompacta, Dokl. Akad. Nauk SSSR 241 (1978), no. 3, 509–512 (Russian). MR 504217
B. Boldjiev and V. Malyhin, The sequentiality is equivalent to the ${\scr F}$-Fréchet-Urysohn property, Comment. Math. Univ. Carolin. 31 (1990), no. 1, 23–25. MR 1056166
S. P. Franklin, Spaces in which sequences suffice. II, Fund. Math. 61 (1967), 51–56. MR 222832, DOI 10.4064/fm-61-1-51-56
Salvador García-Ferreira, On $\textrm {FU}(p)$-spaces and $p$-sequential spaces, Comment. Math. Univ. Carolin. 32 (1991), no. 1, 161–171. MR 1118299
Salvador García-Ferreira and Angel Tamariz-Mascarúa, On $p$-sequential $p$-compact spaces, Comment. Math. Univ. Carolin. 34 (1993), no. 2, 347–356. MR 1241743
Miroslav Katětov, Products of filters, Comment. Math. Univ. Carolinae 9 (1968), 173–189. MR 250257
Ljubiša Kočinac, A generalization of chain-net spaces, Publ. Inst. Math. (Beograd) (N.S.) 44(58) (1988), 109–114. MR 995414
A. P. Kombarov, On a theorem of A. Stone, Dokl. Akad. Nauk SSSR 270 (1983), no. 1, 38–40 (Russian). MR 705191
V. I. Malyhin, Sequential bicompacta: Čech-Stone extensions and $\pi$-points, Vestnik Moskov. Univ. Ser. I Mat. Meh. 30 (1975), no. 1, 23–29 (Russian, with English summary). MR 0375236
V. I. Malyhin, Sequential and Fréchet-Uryson bicompacta, Vestnik Moskov. Univ. Ser. I Mat. Meh. 31 (1976), no. 5, 42–48 (Russian, with English summary). MR 0487955
V. I. Malychin, The sequentiality and the Fréchet-Urysohn property with respect to ultrafilters, Acta Univ. Carolin. Math. Phys. 31 (1990), no. 2, 65–69. 18th Winter School on Abstract Analysis (Srní, 1990). MR 1101417
Ch. Mills, An easier proof of the Shelah $P$-point independence theorem (to appear).
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
A. Szymański, The existence of $P(\alpha )$-points of $N^*$ for $\aleph _{0}<\alpha <{\mathfrak {c}}$, Colloq. Math. 37 (1977), no. 2, 179–184. MR 487992
Edward L. Wimmers, The Shelah $P$-point independence theorem, Israel J. Math. 43 (1982), no. 1, 28–48. MR 728877, DOI 10.1007/BF02761683