EBERLEIN–ŠMULYAN THEOREM FOR ABELIAN TOPOLOGICAL
... fundamental tool, and the optimal situation is when it can be used in its sequential version. Unfortunately this is not always the case, and there is a strong need to look for classes of topological spaces where compactness is equivalent to sequential or countable compactness. It was known from the ...
... fundamental tool, and the optimal situation is when it can be used in its sequential version. Unfortunately this is not always the case, and there is a strong need to look for classes of topological spaces where compactness is equivalent to sequential or countable compactness. It was known from the ...
REVIEW OF GENERAL TOPOLOGY I WOMP 2007 1. Basic Definitions
... Definition 1.7. If T1 and T2 are topologies on a set X, we say the first is finer than (or is a refinement of) the second if T2 ⊆ T1 . Definition 1.8. Let (X, T ) and (Y, S ) be topological spaces, and let f be a function X → Y . We say f is (1) continuous if f −1 (V ) ∈ T for every V ∈ S ; (2) open ...
... Definition 1.7. If T1 and T2 are topologies on a set X, we say the first is finer than (or is a refinement of) the second if T2 ⊆ T1 . Definition 1.8. Let (X, T ) and (Y, S ) be topological spaces, and let f be a function X → Y . We say f is (1) continuous if f −1 (V ) ∈ T for every V ∈ S ; (2) open ...
A NOTE ON PARACOMPACT SPACES
... let X be a regular space satisfying (*) ; we must show that X is paracompact. By Lemma 1, we need only show that every open covering R of X has a locally finite refinement. Now by assumption, R has an open refinement V=ü¡°=1Vi, where each Vi is locally finite. Let Vi be the union of the elements of ...
... let X be a regular space satisfying (*) ; we must show that X is paracompact. By Lemma 1, we need only show that every open covering R of X has a locally finite refinement. Now by assumption, R has an open refinement V=ü¡°=1Vi, where each Vi is locally finite. Let Vi be the union of the elements of ...
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.