| Title:
|
Coherent ultrafilters and nonhomogeneity (English) |
| Author:
|
Starý, Jan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
56 |
| Issue:
|
2 |
| Year:
|
2015 |
| Pages:
|
257-264 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We introduce the notion of a {coherent $P$-ultrafilter} on a complete ccc Boolean algebra, strengthening the notion of a $P$-point on $\omega$, and show that these ultrafilters exist generically under $\mathfrak c = \mathfrak d$. This improves the known existence result of Ketonen [{On the existence of $P$-points in the Stone-Čech compactification of integers}, Fund. Math. {92} (1976), 91--94]. Similarly, the existence theorem of Canjar [{On the generic existence of special ultrafilters}, Proc. Amer. Math. Soc. {110} (1990), no. 1, 233--241] can be extended to show that {coherently selective ultrafilters} exist generically under $\mathfrak c = \operatorname{cov}\mathcal M$. We use these ultrafilters in a topological application: a coherent $P$-ultrafilter on an algebra $\mathcal B$ is an {untouchable point} in the Stone space of $\mathcal B$, witnessing its nonhomogeneity. (English) |
| Keyword:
|
nonhomogeneity |
| Keyword:
|
ultrafilter |
| Keyword:
|
Boolean algebra |
| Keyword:
|
untouchable point |
| MSC:
|
06E10 |
| MSC:
|
54A20 |
| MSC:
|
54G05 |
| idZBL:
|
Zbl 06433823 |
| idMR:
|
MR3338738 |
| DOI:
|
10.14712/1213-7243.2015.123 |
| . |
| Date available:
|
2015-04-25T17:09:59Z |
| Last updated:
|
2017-08-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144246 |
| . |
| Reference:
|
[BS] Balcar B., Simon P.: On minimal $\pi$-character of points in extremally disconnected compact spaces.Topology Appl. 41 (1991), 133–145. Zbl 0752.54013, MR 1129703, 10.1016/0166-8641(91)90105-U |
| Reference:
|
[C] Canjar R.M.: On the generic existence of special ultrafilters.Proc. Amer. Math. Soc. 110 (1990), no. 1, 233–241. Zbl 0715.03018, MR 0993747, 10.1090/S0002-9939-1990-0993747-3 |
| Reference:
|
[F] Frolík Z.: Maps of extremally disconnected spaces, theory of types, and applications.in Franklin, Frolík, Koutník (eds.), General Topology and Its Relations to Modern Analysis and Algebra, Proceedings of the Kanpur topological conference (1971), pp. 131–142. MR 0295305 |
| Reference:
|
[K] Ketonen J.: On the existence of $P$-points in the Stone-Čech compactification of integers.Fund. Math. 92 (1976), 91–94. Zbl 0339.54035, MR 0433387 |
| Reference:
|
[S] Simon P.: Points in extremally disconnected compact spaces.Rend. Circ. Mat. Palermo (2). Suppl. 24 (1990), 203–213. Zbl 0752.54013, MR 1108207 |
| Reference:
|
[W] Wimmer E.L.: The Shelah $P$-point independence theorem.Israel J. Math 43 (1982), no. 1, 28–48. MR 0728877, 10.1007/BF02761683 |
| . |
