BAIRE`S THEOREM AND ITS APPLICATIONS The completeness of
... The symbol δV stands for the set {δy : y ∈ Y }, i.e., the set of all y ∈ Y with k y k< δ. It follows from theorem conditions and the linearity of Ω that the image of every open ball in X, with center x0 , say, contains an open ball in Y with center at Ωx0 . Hence the image of every open set is open. ...
... The symbol δV stands for the set {δy : y ∈ Y }, i.e., the set of all y ∈ Y with k y k< δ. It follows from theorem conditions and the linearity of Ω that the image of every open ball in X, with center x0 , say, contains an open ball in Y with center at Ωx0 . Hence the image of every open set is open. ...
geopolitics of the indian ocean in the post
... and pairwise sg-Lindelöf spaces. Interrelationships between these new concepts and other pairwise covering axioms are established. We also define and study paiwise sg-continuous functions. ...
... and pairwise sg-Lindelöf spaces. Interrelationships between these new concepts and other pairwise covering axioms are established. We also define and study paiwise sg-continuous functions. ...
Some cardinal properties of complete linked systems with compact elements and absolute regular spaces
... E = O (U1 , U2 , . . . , Un ) hV1 , V2 , . . . , Vs i be arbitrary nonempty open base element in Ncm X. Recall that the system S(E) = {Ui ∩ Vj : i = 1, 2, . . . , n, j = 1, 2, . . . , s} ∪ S(O) of open in X subsets is called the pairwise trace of the base element E in X. Here S(O) is the pairwise tr ...
... E = O (U1 , U2 , . . . , Un ) hV1 , V2 , . . . , Vs i be arbitrary nonempty open base element in Ncm X. Recall that the system S(E) = {Ui ∩ Vj : i = 1, 2, . . . , n, j = 1, 2, . . . , s} ∪ S(O) of open in X subsets is called the pairwise trace of the base element E in X. Here S(O) is the pairwise tr ...
Takashi Noiri and Valeriu Popa THE UNIFIED THEORY
... Proof. (1) It is obvious that ∅, X ∈ mω . Let {Aα : α ∈ Λ} be any subfamily of mω . Then for each x ∈ ∪α∈Λ Aα , there exists α0 ∈ Λ such that x ∈ Aα0 . Since Aα0 ∈ mω , there exists Ux ∈ mX containing x such that Ux \ Aα0 is a countable set. Since Ux \(∪α∈Λ Aα ) ⊂ Ux \Aα0 , Ux \(∪α∈Λ Aα ) is a count ...
... Proof. (1) It is obvious that ∅, X ∈ mω . Let {Aα : α ∈ Λ} be any subfamily of mω . Then for each x ∈ ∪α∈Λ Aα , there exists α0 ∈ Λ such that x ∈ Aα0 . Since Aα0 ∈ mω , there exists Ux ∈ mX containing x such that Ux \ Aα0 is a countable set. Since Ux \(∪α∈Λ Aα ) ⊂ Ux \Aα0 , Ux \(∪α∈Λ Aα ) is a count ...
On Noetherian Spaces - LSV
... family (xi )i∈I of elements quasi-ordered by ≤ is a nonempty family such that for every i, j ∈ I there is k ∈ I such that xi ≤ xk and xj ≤ xk .) Write ↑ E = {x ∈ X|∃y ∈ E · y ≤ x}, ↓ E = {x ∈ X|∃y ∈ E · x ≤ y}. If K is compact, then ↑ K is, too, and is also saturated. We shall usually reserve the le ...
... family (xi )i∈I of elements quasi-ordered by ≤ is a nonempty family such that for every i, j ∈ I there is k ∈ I such that xi ≤ xk and xj ≤ xk .) Write ↑ E = {x ∈ X|∃y ∈ E · y ≤ x}, ↓ E = {x ∈ X|∃y ∈ E · x ≤ y}. If K is compact, then ↑ K is, too, and is also saturated. We shall usually reserve the le ...
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... that f ◦ g ' idY (i.e. f ◦ g is homotopic to the identity mapping on Y ), and g ◦ f ' idX , then f is a homotopy equivalence. This homotopy equivalence is sometimes called strong homotopy equivalence to distinguish it from weak homotopy equivalence. If there exist a homotopy equivalence between the ...
... that f ◦ g ' idY (i.e. f ◦ g is homotopic to the identity mapping on Y ), and g ◦ f ' idX , then f is a homotopy equivalence. This homotopy equivalence is sometimes called strong homotopy equivalence to distinguish it from weak homotopy equivalence. If there exist a homotopy equivalence between the ...
3. The Sheaf of Regular Functions
... After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as “nice maps preserving the structure of the variety”. In this chapter we will look at the easiest case of this: the so-called regular functions, i. e. maps to ...
... After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as “nice maps preserving the structure of the variety”. In this chapter we will look at the easiest case of this: the so-called regular functions, i. e. maps to ...
