nLab Wallman compactification

Wallman compactification is a particular compactification of $T_1$-topological spaces introduced in

• Henry Wallman, Lattices and topological spaces, Annals of Math. 39:1 (Jan., 1938), pp. 112-126 jstor

There are recent applications related to topoi and noncommutative geometry

• Olivia Caramello, Gelfand spectra and Wallman compactifications, arxiv/1204.3244; Dualité de Gelfand et bases de Wallman, seminar talk videos

A Wallman base $B$ for a topological space $X$ is a sublattice of of the frame $Open(X)$ of open sets of $X$ which is a base for the topology and satisfies the property that for any $U\in B$ and $x\in U$ there exists $V\in B$ such that $U\cup V = X$ and $x\notin V$.

Standard references are

• P. T. Johnstone, Stone Spaces, Cambridge Studies in Advanced Math. 3 (1982).
• [eom]: Wallman compactification
• wikipedia: Wallman compactification

category: topology