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Difference between revisions of "Interior of a set"

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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518501.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518502.png" />''
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''$X$ in a topological space $Y$''
  
The set of interior points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518503.png" /> (cf. [[Interior point of a set|Interior point of a set]]). It is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518504.png" />. Invariably, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518505.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518506.png" /> is the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518507.png" />. The interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518508.png" /> is also equal to the union of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051850/i0518509.png" /> that are open in the entire space. The interior of a set is sometimes known as the open kernel.
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The set of interior points of $X$ (cf. [[Interior point of a set|Interior point of a set]]). It is usually denoted by $\operatorname{Int}X$. Invariably, $\operatorname{Int}X=X\setminus[Y\setminus X]=X\setminus\operatorname{Fr}x$, where $\operatorname{Fr}$ is the boundary of $X$. The interior of $X$ is also equal to the union of all subsets of $X$ that are open in the entire space. The interior of a set is sometimes known as the open kernel.

Revision as of 17:24, 11 April 2014

$X$ in a topological space $Y$

The set of interior points of $X$ (cf. Interior point of a set). It is usually denoted by $\operatorname{Int}X$. Invariably, $\operatorname{Int}X=X\setminus[Y\setminus X]=X\setminus\operatorname{Fr}x$, where $\operatorname{Fr}$ is the boundary of $X$. The interior of $X$ is also equal to the union of all subsets of $X$ that are open in the entire space. The interior of a set is sometimes known as the open kernel.

How to Cite This Entry:
Interior of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interior_of_a_set&oldid=31529
This article was adapted from an original article by S.M. Sirota (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article