“Quasi-uniform spaces”
... d(y, x) whenever x, y ∈ X is the conjugate quasi-pseudometric of d. A quasipseudometric d on X is called a quasi-metric if x, y ∈ X and d(x, y) = 0 imply x = y; it is called non-archimedean if d(x, z) ≤ max{d(x, y), d(y, z)} whenever x, y, z ∈ X. Each quasi-pseudometric d on X generates a quasi-unif ...
... d(y, x) whenever x, y ∈ X is the conjugate quasi-pseudometric of d. A quasipseudometric d on X is called a quasi-metric if x, y ∈ X and d(x, y) = 0 imply x = y; it is called non-archimedean if d(x, z) ≤ max{d(x, y), d(y, z)} whenever x, y, z ∈ X. Each quasi-pseudometric d on X generates a quasi-unif ...
For the Oral Candidacy examination, the student is examined in
... For the Oral Candidacy examination, the student is examined in three basic subjects (satisfying the requirements of the student's intended Track of Specialization). For each subject, the student must master all of the topics listed on the syllabus. The student is expected to have a through understan ...
... For the Oral Candidacy examination, the student is examined in three basic subjects (satisfying the requirements of the student's intended Track of Specialization). For each subject, the student must master all of the topics listed on the syllabus. The student is expected to have a through understan ...
Metric and Banach spaces
... Theorem B.2 Let (X, dX ) and (Y, dY ) be two metric spaces and let consider a uniformely continuous function f : (X, dX ) → (Y, dY ). If (xn )n∈N is a Cauchy sequence of X, then f (xn )n∈N is a Cauchy sequence of F . The reciprocal one is not true. Proposition B.6 We have two properties about conver ...
... Theorem B.2 Let (X, dX ) and (Y, dY ) be two metric spaces and let consider a uniformely continuous function f : (X, dX ) → (Y, dY ). If (xn )n∈N is a Cauchy sequence of X, then f (xn )n∈N is a Cauchy sequence of F . The reciprocal one is not true. Proposition B.6 We have two properties about conver ...
Course 421: Algebraic Topology Section 1
... product topology) if, given any point p of U , there exist open sets Vi in Xi for i = 1, 2, . . . , n such that {p} ⊂ V1 × V2 × · · · × Vn ⊂ U . Lemma 1.8 Let X1 , X2 , . . . , Xn be topological spaces. Then the collection of open sets in X1 × X2 × · · · × Xn is a topology on X1 × X2 × · · · × Xn . ...
... product topology) if, given any point p of U , there exist open sets Vi in Xi for i = 1, 2, . . . , n such that {p} ⊂ V1 × V2 × · · · × Vn ⊂ U . Lemma 1.8 Let X1 , X2 , . . . , Xn be topological spaces. Then the collection of open sets in X1 × X2 × · · · × Xn is a topology on X1 × X2 × · · · × Xn . ...
Contra-e-Continuous Functions 1 Introduction
... Proof. The “if” part is easy to prove. To prove “only if” part, let g ◦ f : X → Z is contra-e-continuous and let F be a closed subset of Z. Then (g ◦ f )−1 (F ) is an e-open subset of X i.e. f −1 (g −1 (F )) is pre-e-open in X. Since f is e-open, f (f −1 (g −1 (F ))) is an e-open subset of Y and so ...
... Proof. The “if” part is easy to prove. To prove “only if” part, let g ◦ f : X → Z is contra-e-continuous and let F be a closed subset of Z. Then (g ◦ f )−1 (F ) is an e-open subset of X i.e. f −1 (g −1 (F )) is pre-e-open in X. Since f is e-open, f (f −1 (g −1 (F ))) is an e-open subset of Y and so ...
