Sober Spaces, Well-Filtration and Compactness Principles
... means that the power sets are the only ∪-irreducible systems of finite character. Indeed, the Scott-closed subsets of power sets are just the systems of finite character (containing a set if and only if all of its finite subsets belong to the system). Assume now that X S is a system of finite charac ...
... means that the power sets are the only ∪-irreducible systems of finite character. Indeed, the Scott-closed subsets of power sets are just the systems of finite character (containing a set if and only if all of its finite subsets belong to the system). Assume now that X S is a system of finite charac ...
The Closed Limit Point Compactness
... = , ∅, { , }, { , } . The sets = { , } and = { , } are closed in and has limit point i.e. = and = . Hence is a closed limit point compact. (ii) The real numbers with the usual topology R , is not a closed limit point compact. {3} is non-empty proper closed set in R which has no a limit point compact ...
... = , ∅, { , }, { , } . The sets = { , } and = { , } are closed in and has limit point i.e. = and = . Hence is a closed limit point compact. (ii) The real numbers with the usual topology R , is not a closed limit point compact. {3} is non-empty proper closed set in R which has no a limit point compact ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.