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As defined in [[#References|[a1]]], a point in a [[Completely-regular space|completely-regular space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p1300102.png" /> at which any [[Prime ideal|prime ideal]] of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p1300103.png" /> of real-valued continuous functions is maximal (cf. also [[Continuous function|Continuous function]]; [[Maximal ideal|Maximal ideal]]). A prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p1300104.png" /> is  "at x"  if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p1300106.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p1300107.png" />; thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p1300108.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p1300109.png" />-point if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001010.png" /> is the only prime ideal at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001011.png" />. Equivalent formulations are:
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As defined in [[#References|[a1]]], a point in a [[completely-regular space]] $X$ at which any [[prime ideal]] of the ring $C(X)$ of real-valued [[continuous function]]s is maximal. A prime ideal $\mathfrak{P}$ is  "at $x$"  if $f(x)=0$ for all $f \in \mathfrak{P}$; thus $x$ is a $P$-point if and only if $\mathfrak{M}_x = \{f \in C(X) : f(x) = 0 \}$ is the only prime ideal at $x$. Equivalent formulations are:
  
1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001012.png" /> is a continuous function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001014.png" /> vanishes on a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001015.png" />; and
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1) if $f$ is a continuous function and $f(x)=0$, then $f$ vanishes on a neighbourhood of $x$; and
  
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.
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2) every countable intersection of neighbourhoods of $x$ contains a neighbourhood of $x$. The latter is commonly used to define $P$-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.
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Of particular interest are $P$-points in the space $\mathbf{N}^* = \beta\mathbf{N}\setminus\mathbf{N}$, 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 [[#References|[a2]]] proved that the space $\mathbf{N}^*$ has $P$-points if the [[continuum hypothesis]] is assumed; this showed that $\mathbf{N}^*$ cannot be proved homogeneous (cf. also [[Homogeneous space]]), because not every point in an infinite compact space can be a $P$-point. Points of $\mathbf{N}^*$ are identified with free [[ultrafilter]]s on the set $\mathbf{N}$. A point or ultrafilter $u$ is a $P$-point if and only if for every sequence $(U_n)$ of elements of $u$ there is an element $U$ of $u$ such that $U \stackrel{*}{\subseteq} U_n$ for all $n$, where $A \stackrel{*}{\subseteq} B$ means that $A\setminus B$ is finite. Equivalently, $u$ is a $P$-point if and only if for every partition $\{A_n\}$ of $\mathbf{N}$ either there is an $n$ such that $A_n \in u$ or there is a $U\in u$ such that $U \cap A_n$ is finite for all $n$. S. Shelah [[#References|[a3]]] constructed a model of set theory in which $\mathbf{N}^*$ has no $P$-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" />).
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There is continued interest in $P$-point ultrafilters because of their combinatorial properties; e.g., $u$ is a $P$-point if and only if for every function $f : \mathbf{N} \rightarrow \mathbf{R}$ there is an element $U$ of $u$ such that $f[U]$ is a converging sequence (possibly to $\infty$ or $-\infty$).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Gillman,  M. Henriksen,  "Concerning rings of continuous functions"  ''Trans. Amer. Math. Soc.'' , '''77'''  (1954)  pp. 340–362</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Homogeneity problems in the theory of Čech compactifications"  ''Duke Math. J.'' , '''23'''  (1956)  pp. 409–419; 633</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Wimmers,  "The Shelah <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130010/p13001059.png" />-point independence theorem"  ''Israel J. Math.'' , '''43'''  (1982)  pp. 28–48</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Gillman,  M. Henriksen,  "Concerning rings of continuous functions"  ''Trans. Amer. Math. Soc.'' , '''77'''  (1954)  pp. 340–362</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Homogeneity problems in the theory of Čech compactifications"  ''Duke Math. J.'' , '''23'''  (1956)  pp. 409–419; 633</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Wimmers,  "The Shelah $P$-point independence theorem"  ''Israel J. Math.'' , '''43'''  (1982)  pp. 28–48</TD></TR>
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</table>
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 +
{{TEX|done}}

Latest revision as of 05:16, 21 November 2017

As defined in [a1], a point in a completely-regular space $X$ at which any prime ideal of the ring $C(X)$ of real-valued continuous functions is maximal. A prime ideal $\mathfrak{P}$ is "at $x$" if $f(x)=0$ for all $f \in \mathfrak{P}$; thus $x$ is a $P$-point if and only if $\mathfrak{M}_x = \{f \in C(X) : f(x) = 0 \}$ is the only prime ideal at $x$. Equivalent formulations are:

1) if $f$ is a continuous function and $f(x)=0$, then $f$ vanishes on a neighbourhood of $x$; and

2) every countable intersection of neighbourhoods of $x$ contains a neighbourhood of $x$. The latter is commonly used to define $P$-points in arbitrary topological spaces.

Of particular interest are $P$-points in the space $\mathbf{N}^* = \beta\mathbf{N}\setminus\mathbf{N}$, 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 $\mathbf{N}^*$ has $P$-points if the continuum hypothesis is assumed; this showed that $\mathbf{N}^*$ cannot be proved homogeneous (cf. also Homogeneous space), because not every point in an infinite compact space can be a $P$-point. Points of $\mathbf{N}^*$ are identified with free ultrafilters on the set $\mathbf{N}$. A point or ultrafilter $u$ is a $P$-point if and only if for every sequence $(U_n)$ of elements of $u$ there is an element $U$ of $u$ such that $U \stackrel{*}{\subseteq} U_n$ for all $n$, where $A \stackrel{*}{\subseteq} B$ means that $A\setminus B$ is finite. Equivalently, $u$ is a $P$-point if and only if for every partition $\{A_n\}$ of $\mathbf{N}$ either there is an $n$ such that $A_n \in u$ or there is a $U\in u$ such that $U \cap A_n$ is finite for all $n$. S. Shelah [a3] constructed a model of set theory in which $\mathbf{N}^*$ has no $P$-points, thus showing that Rudin's theorem is not definitive.

There is continued interest in $P$-point ultrafilters because of their combinatorial properties; e.g., $u$ is a $P$-point if and only if for every function $f : \mathbf{N} \rightarrow \mathbf{R}$ there is an element $U$ of $u$ such that $f[U]$ is a converging sequence (possibly to $\infty$ or $-\infty$).

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 $P$-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
This article was adapted from an original article by K.P. Hart (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article