View source for Wallman compactification

{{MSC|54D35}}

''Wallman–Shanin compactification, $\omega X$, of a topological space $X$ satisfying the [[separation axiom]] $T_1$''

The space whose points are maximal [[Centred family of sets|centred systems]] of closed sets $\xi = \{F_\alpha \}$ in $X$. The topology in $\omega X$ is given by the [[closed base]] $\{ \Phi_F \}$, where $F$ ranges over all closed sets in $X$ and $\Phi_F$ consists of precisely those $\xi = \{F_\alpha \}$ for which $F = F_\alpha$ for some $\alpha$.

This compactification was described by H. Wallman [[#References|[1]]].

The Wallman compactification is always a compact $T_1$-space; for a normal space it coincides with the [[Stone–Čech compactification]].

If in defining the extension $\omega X$ one chooses not all closed sets, but only those contained in a certain fixed closed base, one obtains a so-called compactification of Wallman type. Not every Hausdorff compactification of a [[Tikhonov space]] is a compactification of Wallman type.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  H. Wallman,   "Lattices and topological spaces"  ''Ann of Math.'' , '''39'''  (1938)  pp. 112–126</TD></TR>
</table>



====Comments====
Compactifications that are not Wallman compactifications were constructed by V.M. Ul'yanov [[#References|[a1]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  V.M. Ul'yanov,   "Solution of a basic problem on compactifications of Wallman type"  ''Soviet Math. Dokl.'' , '''18'''  (1977)  pp. 567–571  ''Dokl. Akad. Nauk SSSR'' , '''233''' :  6  (1977)  pp. 1056–1059</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.A. Alo,   H.L. Shapiro,   "Normal bases and compactifications"  ''Math. Ann.'' , '''175'''  (1968)  pp. 337–340</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Frink,   "Compactifications and semi-normal spaces"  ''Amer. J. Math.'' , '''86'''  (1964)  pp. 602–607</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.C. Walker,   "The Stone–Čech compactification" , Springer  (1974)</TD></TR>
</table>

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