3. Sheaves of groups and rings.
... (3.1) Definition. Let X be a topological space and let B be a basis for the topology. Moreover let !F! and !A! be presheaves defined on B. We say that F takes values in groups, or is a presheaf of groups, on B if for every subset U of X belonging to B we have that F(U ) is a group, and for every inc ...
... (3.1) Definition. Let X be a topological space and let B be a basis for the topology. Moreover let !F! and !A! be presheaves defined on B. We say that F takes values in groups, or is a presheaf of groups, on B if for every subset U of X belonging to B we have that F(U ) is a group, and for every inc ...
279 ASCOLI`S THEOREM IN ALMOST QUIET QUASI
... Definition 1.6. [7] A set A ⊂ (X, τ ) is said to be N-closed in X or simply N-closed, if for any cover of A by τ -open sets, there exists a finite subcollection the interiors of the closures of which cover A; interiors and closures are of course w.r.t. τ . A set (X, τ ) is said to be nearly compact ...
... Definition 1.6. [7] A set A ⊂ (X, τ ) is said to be N-closed in X or simply N-closed, if for any cover of A by τ -open sets, there exists a finite subcollection the interiors of the closures of which cover A; interiors and closures are of course w.r.t. τ . A set (X, τ ) is said to be nearly compact ...
A1 Partitions of unity
... closure. If U is a countable basis for the topology, it is easy to see that those U ∈ U that are contained in some Vx are still a basis, and that U is compact for all such U . These sets U are a countable collection of compact sets whose union is X. ¤ Exercise A1.4 shows that a topological space tha ...
... closure. If U is a countable basis for the topology, it is easy to see that those U ∈ U that are contained in some Vx are still a basis, and that U is compact for all such U . These sets U are a countable collection of compact sets whose union is X. ¤ Exercise A1.4 shows that a topological space tha ...
1 Preliminaries
... A topological space is irreducible if any (and hence all) of the following conditions hold: • The space is not the union of two proper non-empty closed sets • Any two non-empty open sets intersect • Any non-empty open set is dense A subset of a topological space irreducible if it is irreducible with ...
... A topological space is irreducible if any (and hence all) of the following conditions hold: • The space is not the union of two proper non-empty closed sets • Any two non-empty open sets intersect • Any non-empty open set is dense A subset of a topological space irreducible if it is irreducible with ...
Finite Topological Spaces - Trace: Tennessee Research and
... Theorem 3.1. Let (X, T ) and (Y, Γ) be topological spaces where X is connected. If f : X → Y is continuous then f (X) is connected. Proof. Suppose to the contrary that {U, V } is a separation of f (X) = Z. Then U and V are each open in the subspace topology of Z. Hence U = Z ∩ Uz and V = Z ∩ Vz wher ...
... Theorem 3.1. Let (X, T ) and (Y, Γ) be topological spaces where X is connected. If f : X → Y is continuous then f (X) is connected. Proof. Suppose to the contrary that {U, V } is a separation of f (X) = Z. Then U and V are each open in the subspace topology of Z. Hence U = Z ∩ Uz and V = Z ∩ Vz wher ...
