Difference between revisions of "P-point"
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2) every countable intersection of neighbourhoods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001016.png" /> contains a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001017.png" />. The latter is commonly used to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001018.png" />-points in arbitrary topological spaces. | 2) every countable intersection of neighbourhoods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001016.png" /> contains a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001017.png" />. The latter is commonly used to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001018.png" />-points in arbitrary topological spaces. | ||
| − | Of particular interest are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001019.png" />-points in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001020.png" />, the remainder in the [[Stone–Čech compactification|Stone–Čech compactification]] of the space of natural numbers. This is so because W. Rudin [[#References|[a2]]] proved that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001021.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001022.png" />-points if the [[Continuum hypothesis|continuum hypothesis]] is assumed; this showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001023.png" /> cannot be proved homogeneous (cf. also [[Homogeneous space|Homogeneous space]]), because not every point in an infinite compact space can be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001024.png" />-point. Points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001025.png" /> are identified with free ultrafilters on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001026.png" /> (cf. also [[Ultrafilter|Ultrafilter]]). A point or ultrafilter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001027.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001028.png" />-point if and only if for every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001029.png" /> of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001030.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001035.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001036.png" /> is finite. Equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001037.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001038.png" />-point if and only if for every partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001040.png" /> either there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001041.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001042.png" /> or there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001044.png" /> is finite for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001045.png" />. S. Shelah [[#References|[a3]]] constructed a model of set theory in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001046.png" /> has no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001047.png" />-points, thus showing that Rudin's theorem is not definitive. | + | Of particular interest are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001019.png" />-points in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001020.png" />, the remainder in the [[Stone–Čech compactification|Stone–Čech compactification]] of the space of natural numbers (cf. [[Cech-Stone compactification of omega]]). This is so because W. Rudin [[#References|[a2]]] proved that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001021.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001022.png" />-points if the [[Continuum hypothesis|continuum hypothesis]] is assumed; this showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001023.png" /> cannot be proved homogeneous (cf. also [[Homogeneous space|Homogeneous space]]), because not every point in an infinite compact space can be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001024.png" />-point. Points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001025.png" /> are identified with free ultrafilters on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001026.png" /> (cf. also [[Ultrafilter|Ultrafilter]]). A point or ultrafilter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001027.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001028.png" />-point if and only if for every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001029.png" /> of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001030.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001035.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001036.png" /> is finite. Equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001037.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001038.png" />-point if and only if for every partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001040.png" /> either there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001041.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001042.png" /> or there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001044.png" /> is finite for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001045.png" />. S. Shelah [[#References|[a3]]] constructed a model of set theory in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001046.png" /> has no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001047.png" />-points, thus showing that Rudin's theorem is not definitive. |
There is continued interest in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001050.png" />-point ultrafilters because of their combinatorial properties; e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001051.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001052.png" />-point if and only if for every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001053.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001056.png" /> is a converging sequence (possibly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001057.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001058.png" />). | There is continued interest in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001050.png" />-point ultrafilters because of their combinatorial properties; e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001051.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001052.png" />-point if and only if for every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001053.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001056.png" /> is a converging sequence (possibly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001057.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001058.png" />). | ||
Revision as of 21:27, 19 November 2017
As defined in [a1], a point in a completely-regular space 
at which any prime ideal of the ring 
of real-valued continuous functions is maximal (cf. also Continuous function; Maximal ideal). A prime ideal 
is "at x" if 
for all 
; thus 
is a 
-point if and only if 
is the only prime ideal at 
. Equivalent formulations are:
1) if 
is a continuous function and 
, then 
vanishes on a neighbourhood of 
; and
2) every countable intersection of neighbourhoods of 
contains a neighbourhood of 
. The latter is commonly used to define 
-points in arbitrary topological spaces.
Of particular interest are 
-points in the space 
, the remainder in the Stone–Čech compactification of the space of natural numbers (cf. Cech-Stone compactification of omega). This is so because W. Rudin [a2] proved that the space 
has 
-points if the continuum hypothesis is assumed; this showed that 
cannot be proved homogeneous (cf. also Homogeneous space), because not every point in an infinite compact space can be a 
-point. Points of 
are identified with free ultrafilters on the set 
(cf. also Ultrafilter). A point or ultrafilter 
is a 
-point if and only if for every sequence 
of elements of 
there is an element 
of 
such that 
for all 
, where 
means that 
is finite. Equivalently, 
is a 
-point if and only if for every partition 
of 
either there is an 
such that 
or there is a 
such that 
is finite for all 
. S. Shelah [a3] constructed a model of set theory in which 
has no 
-points, thus showing that Rudin's theorem is not definitive.
There is continued interest in 
-point ultrafilters because of their combinatorial properties; e.g., 
is a 
-point if and only if for every function 
there is an element 
of 
such that 
is a converging sequence (possibly to 
or 
).
References
| [a1] | L. Gillman, M. Henriksen, "Concerning rings of continuous functions" Trans. Amer. Math. Soc. , 77 (1954) pp. 340–362 |
| [a2] | W. Rudin, "Homogeneity problems in the theory of Čech compactifications" Duke Math. J. , 23 (1956) pp. 409–419; 633 |
| [a3] | E. Wimmers, "The Shelah  -point independence theorem" Israel J. Math. , 43 (1982) pp. 28–48 |
How to Cite This Entry:
P-point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-point&oldid=15552
P-point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-point&oldid=15552
This article was adapted from an original article by K.P. Hart (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article