Wallman compactification
Encyclopedia
In mathematics, the Wallman compactification is a compactification
of T1 topological spaces
that was constructed by .
The points of the Wallman compactification ωX of a space X are the maximal families Φ of closed nonempty subsets of X such that Φ is closed under finite intersections. A base for the closed sets is given by the families ΦF containing a fixed closed set F of X.
For normal space
s, the Wallman compactification is essentially the same as the Stone–Čech compactification
.
Compactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
of T1 topological spaces
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
that was constructed by .
The points of the Wallman compactification ωX of a space X are the maximal families Φ of closed nonempty subsets of X such that Φ is closed under finite intersections. A base for the closed sets is given by the families ΦF containing a fixed closed set F of X.
For normal space
Normal space
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
s, the Wallman compactification is essentially the same as the Stone–Čech compactification
Stone–Cech compactification
In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX...
.