A totally disconnected T2 space with this property will be isomorphic to a Boolean algebra according to (*) Stone's Representation theorem.
If the set X is a particular point topological set, then it will possess the "pseudo-" form of this property, meaning that the set's image under any continuous function to the reals is bounded. However, set X will lack the "meta-" form of this property, whose definition involves a point finite refinement. Applying an ultrafilter to a topological set with this property will always yield one point. A totally disconnected T2 space with this property will be isomorphic to a Boolean algebra according to (*) Stone's Representation theorem. For a weak topology, the Eberlein-Schmulian theorem states that three different forms of this property are equivalent in a Banach space. The "weakly countably" form of this property holds for a set X if every subset of X has a limit point; another definition of this property is that every open cover of the set has a finite subcover. For 10 points, name this property of a Euclidean topological space making it both closed and bounded.
If the set X is a particular point topological set, then it will possess the "pseudo-" form of this property, meaning that the set's image under any continuous function to the reals is bounded.
compactness