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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b0170201.png" /> in a compactification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b0170202.png" />''
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''$X$ in a compactification $bX$''
  
A finite family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b0170203.png" /> of sets open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b0170204.png" /> such that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b0170205.png" /> is compact, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b0170206.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b0170207.png" /> is the largest open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b0170208.png" /> the intersection of which with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b0170209.png" /> is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702011.png" /> is assumed to be completely regular). The concept of a bordering of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702012.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702013.png" /> coincides with the concept of an almost-extendable bordering of a proximity space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702014.png" /> (the proximity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702015.png" /> is induced by the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702016.png" />), formulated in terms of the proximity: apart from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702017.png" /> being compact, it is necessary that for any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702018.png" />, the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702019.png" /> is a uniform covering of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702020.png" />. A bordering of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702021.png" /> in its Stone–Čech compactification is simply called a bordering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702022.png" />. In the language of borderings, a series of theorems has been formulated on the dimensions of the remainder of compactifications of topological and proximity spaces.
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A finite family $\{U_1,\ldots,U_k\}$ of sets open in $X$ such that the set $K = X \ (U_1 \cup \cdots \cup U_k)$ is compact, and $bX = K \cup \tilde U_1 \cup \cdots \cup \tilde U_k$, where $\tilde U_i$ is the largest open set in $bX$ the intersection of which with $X$ is the set $U_i$ ($X$ is assumed to be [[Completely-regular space|completely regular]]). The concept of a bordering of a space $X$ in $bX$ coincides with the concept of an almost-extendable bordering of a proximity space $X$ (the proximity on $X$ is induced by the extension $bX$), formulated in terms of the proximity: apart from $K$ being compact, it is necessary that for any neighbourhood $O_K$, the family $\{O_k,U_1,\ldots,U_k\}$ is a uniform covering of the space $X$. A bordering of a space $X$ in its [[Stone–Čech compactification]] is simply called a bordering of $X$. In the language of borderings, a series of theorems has been formulated on the dimensions of the remainder of compactifications of topological and proximity spaces.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.M. Smirnov,  "On the dimensions of remainders of compactifications of proximity and topological spaces"  ''Mat. Sb.'' , '''71''' :  4  (1966)  pp. 554–482  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.M. Smirnov,  "On the dimensions of remainders of compactifications of proximity and topological spaces"  ''Mat. Sb.'' , '''71''' :  4  (1966)  pp. 554–482  (In Russian)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
A concept related to the bordering of a space is that of a border cover: A collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702023.png" /> of open sets such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017020/b01702024.png" /> is compact. Border covers work in a sense opposite to borderings. In the case of borderings a compactification is given; from certain systems of border covers one can construct compactifications whose remainders can have special properties.
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A concept related to the bordering of a space is that of a border cover: A collection $\mathcal{U}$ of open sets such that $X \setminus \cup \mathcal{U}$ is compact. Border covers work in a sense opposite to borderings. In the case of borderings a compactification is given; from certain systems of border covers one can construct compactifications whose remainders can have special properties.
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Revision as of 16:00, 24 September 2017

$X$ in a compactification $bX$

A finite family $\{U_1,\ldots,U_k\}$ of sets open in $X$ such that the set $K = X \ (U_1 \cup \cdots \cup U_k)$ is compact, and $bX = K \cup \tilde U_1 \cup \cdots \cup \tilde U_k$, where $\tilde U_i$ is the largest open set in $bX$ the intersection of which with $X$ is the set $U_i$ ($X$ is assumed to be completely regular). The concept of a bordering of a space $X$ in $bX$ coincides with the concept of an almost-extendable bordering of a proximity space $X$ (the proximity on $X$ is induced by the extension $bX$), formulated in terms of the proximity: apart from $K$ being compact, it is necessary that for any neighbourhood $O_K$, the family $\{O_k,U_1,\ldots,U_k\}$ is a uniform covering of the space $X$. A bordering of a space $X$ in its Stone–Čech compactification is simply called a bordering of $X$. In the language of borderings, a series of theorems has been formulated on the dimensions of the remainder of compactifications of topological and proximity spaces.

References

[1] Yu.M. Smirnov, "On the dimensions of remainders of compactifications of proximity and topological spaces" Mat. Sb. , 71 : 4 (1966) pp. 554–482 (In Russian)


Comments

A concept related to the bordering of a space is that of a border cover: A collection $\mathcal{U}$ of open sets such that $X \setminus \cup \mathcal{U}$ is compact. Border covers work in a sense opposite to borderings. In the case of borderings a compactification is given; from certain systems of border covers one can construct compactifications whose remainders can have special properties.

How to Cite This Entry:
Bordering of a space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bordering_of_a_space&oldid=41952
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article