View source for Closure of a set

''in a topological space''

The intersection of all closed sets (cf. [[Closed set|Closed set]]) containing the set.



====Comments====
In the Russian literature the closure of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c0226301.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c0226302.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c0226303.png" /> to express that the closure is taken in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c0226304.png" />, in the Western literature one uses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c0226305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c0226306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c0226307.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c0226308.png" />.

Another definition of closure is as follows. The closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c0226309.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c02263010.png" /> is the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c02263011.png" /> satisfying: Every neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c02263012.png" /> intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c02263013.png" />.

The closure operation satisfies: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c02263014.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c02263015.png" />; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c02263016.png" />; and 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022630/c02263017.png" />. Any operation satisfying 1), 2), 3), and 4) is called a closure operation. One can define a topological space by means of a closure operation: The closed sets are to be those sets that equal their own closure.

This approach is taken in [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Kuratowski,   "Topology" , '''1''' , PWN &amp; Acad. Press  (1966)  (Translated from French)</TD></TR></table>