Sober Spaces, Well-Filtration and Compactness Principles
... F ∈ U ⇔ F ⊆ U for all finite subsets F of L, and that an element p is (∧-)prime iff {a ∈ L : a 6≤ p} is a filter. The Separation Lemma for Quantales or Locales often provides very short proofs of algebraic, topological or order-theoretical statements whose derivation from AC would be more complicate ...
... F ∈ U ⇔ F ⊆ U for all finite subsets F of L, and that an element p is (∧-)prime iff {a ∈ L : a 6≤ p} is a filter. The Separation Lemma for Quantales or Locales often provides very short proofs of algebraic, topological or order-theoretical statements whose derivation from AC would be more complicate ...
A Note on Local Compactness
... by an arbitrary category X which comes equipped with a proper factorization system and a closure operator, as in [4], [5], [3]. The only sticky point at this level of generality is the pullback behaviour of c-open maps (w.r.t. the closure operator c), which is not as smooth as in T op or Loc, but wh ...
... by an arbitrary category X which comes equipped with a proper factorization system and a closure operator, as in [4], [5], [3]. The only sticky point at this level of generality is the pullback behaviour of c-open maps (w.r.t. the closure operator c), which is not as smooth as in T op or Loc, but wh ...
QUOTIENT SPACES – MATH 446 Marc Culler
... Thus the open sets in the quotient topology are collections of subsets whose union is open in X. We can think of the partition elements as “fat points”, and the open sets as collections of “fat points” whose union is open as a subset of X. Example 2.2. For a non-negative integer n, consider the subs ...
... Thus the open sets in the quotient topology are collections of subsets whose union is open in X. We can think of the partition elements as “fat points”, and the open sets as collections of “fat points” whose union is open as a subset of X. Example 2.2. For a non-negative integer n, consider the subs ...
On Hausdorff compactifications - Mathematical Sciences Publishers
... Given a pair of spaces X and Y, a necessary and sufficient condition is found for Y to be homeomorphic to daχ(aX— X) for some compactification aX of X. From this follows a necessary and sufficient condition for Y to be homeomorphic to aX — X for some aX. As an application, a sufficient condition is ...
... Given a pair of spaces X and Y, a necessary and sufficient condition is found for Y to be homeomorphic to daχ(aX— X) for some compactification aX of X. From this follows a necessary and sufficient condition for Y to be homeomorphic to aX — X for some aX. As an application, a sufficient condition is ...
Completely N-continuous Multifunctions
... Theorem 12. Let F : X → Y be a multifunction from a topological space (X, τ ) to a topological space (Y, υ) and let F (X) be endowed with subspace topology. If F is upper completely N-continuous, then F : X → F (X) is upper completely N-continuous. Proof. Since F is upper completely N-continuous, F ...
... Theorem 12. Let F : X → Y be a multifunction from a topological space (X, τ ) to a topological space (Y, υ) and let F (X) be endowed with subspace topology. If F is upper completely N-continuous, then F : X → F (X) is upper completely N-continuous. Proof. Since F is upper completely N-continuous, F ...
Mumford`s conjecture - University of Oxford
... several constructions. One of these comes from complex analysis via Teichmüller spaces. We will discuss this in more detail below as it relates easily to the topological point of view. But it was Mumford who introduced Mg into algebraic geometry. A precise definition and construction of the coarse ...
... several constructions. One of these comes from complex analysis via Teichmüller spaces. We will discuss this in more detail below as it relates easily to the topological point of view. But it was Mumford who introduced Mg into algebraic geometry. A precise definition and construction of the coarse ...
On sigma-Induced L-Fuzzy Topological Spaces
... Since the intersection of two regular -subsets is regular -subset, then the family of all regular -subsets in ( X,T ) forms a base for a smaller topology Tσ on X, called the -semi-regularization of T. A topological space( X,T ) is sad to be -semi-regular iff T = Tσ .i.e. (X, Tσ) space is the -semire ...
... Since the intersection of two regular -subsets is regular -subset, then the family of all regular -subsets in ( X,T ) forms a base for a smaller topology Tσ on X, called the -semi-regularization of T. A topological space( X,T ) is sad to be -semi-regular iff T = Tσ .i.e. (X, Tσ) space is the -semire ...
