|
|
| (4 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| − | The set of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i0518101.png" /> of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i0518102.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i0518103.png" /> for which an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i0518104.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i0518105.png" /> exists such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i0518106.png" />. The interior of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i0518107.png" /> is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i0518108.png" /> and represents the largest open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i0518109.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i05181010.png" />. The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i05181011.png" /> holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i05181012.png" /> denotes closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i05181013.png" />. The interior of a set in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051810/i05181014.png" /> is a regular open or [[Canonical set|canonical set]]. Spaces in which the open canonical sets form a [[Base|base]] for the topology are called semi-regular. Every regular space is semi-regular. The interior is sometimes called the open kernel of the set.
| + | #REDIRECT [[Interior of a set]] |
| − | | |
| − | | |
| − | | |
| − | ====Comments====
| |
| − | See also [[Interior of a set|Interior of a set]].
| |
Latest revision as of 20:14, 19 December 2015
How to Cite This Entry:
Interior. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interior&oldid=13174
This article was adapted from an original article by V.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article