Wallman compactification

2020 Mathematics Subject Classification: Primary: 54D35 [MSN][ZBL]

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

The space whose points are maximal 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 [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

[1]	H. Wallman, "Lattices and topological spaces" Ann of Math. , 39 (1938) pp. 112–126

Comments

Compactifications that are not Wallman compactifications were constructed by V.M. Ul'yanov [a1].

References

[a1]	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
[a2]	R.A. Alo, H.L. Shapiro, "Normal bases and compactifications" Math. Ann. , 175 (1968) pp. 337–340
[a3]	O. Frink, "Compactifications and semi-normal spaces" Amer. J. Math. , 86 (1964) pp. 602–607
[a4]	R.C. Walker, "The Stone–Čech compactification" , Springer (1974)