SOME RESULTS ABOUT BANACH COMPACT ALGEBRAS B. M.
... University of the North, Private Bag X1106, Sovenga, 0727, South Africa ...
... University of the North, Private Bag X1106, Sovenga, 0727, South Africa ...
THE CLOSED-POINT ZARISKI TOPOLOGY FOR
... it is easily seen [4, 2.5(i)] that this topology is a refinement of the natural Zariski topology, in which the closed sets all have the form V (I) : = {[N ] ∈ Irr R | I.N = 0} , where I is an ideal of R, and where [N ] denotes the isomorphism class of a simple module N . Our aim is to show that, und ...
... it is easily seen [4, 2.5(i)] that this topology is a refinement of the natural Zariski topology, in which the closed sets all have the form V (I) : = {[N ] ∈ Irr R | I.N = 0} , where I is an ideal of R, and where [N ] denotes the isomorphism class of a simple module N . Our aim is to show that, und ...
On ωb-open sets and b
... Definition 3.12. A function f : X → Y is said to be ωb-closed if f (A) ωb-closed in Y for each b-closed set A of X . Theorem 3.13. If f : X → Y is an ωb-closed surjection such that f −1 ( y) is b-Lindelöf relative to X and Y is b-Lindelöf, then X is b-Lindelöf. Proof. Let {Uα : α ∈ Λ} be any b-open ...
... Definition 3.12. A function f : X → Y is said to be ωb-closed if f (A) ωb-closed in Y for each b-closed set A of X . Theorem 3.13. If f : X → Y is an ωb-closed surjection such that f −1 ( y) is b-Lindelöf relative to X and Y is b-Lindelöf, then X is b-Lindelöf. Proof. Let {Uα : α ∈ Λ} be any b-open ...
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 ...
... Here f : (X,τ1, G1) → (Y,τ2, G2) is a almost homeomorphism mapping of an G1-NC space X on to Y. Let U ={Uα : α ∈ Λ} be any regular open cover of Y. Then f being almost continuous, U* ={f −1(Uα) : α ∈ Λ} is an open cover of the G1-NC space X. Therefore there exists a finite subfamily, {f −1(Uα ) : i ...
For printing
... directed system Δ with values in Y, and is denoted by \y \ ^ or more briefly by XJμ] 5 a directed set [yμ] converges to y (in symbols, γμ —» y) if for every neighborhood V of yj^here exists a μ' e Δ with y e F for all μ > μ'. Furthermore, we have g: Y —» Z if and only if for every directed s e t { y ...
... directed system Δ with values in Y, and is denoted by \y \ ^ or more briefly by XJμ] 5 a directed set [yμ] converges to y (in symbols, γμ —» y) if for every neighborhood V of yj^here exists a μ' e Δ with y e F for all μ > μ'. Furthermore, we have g: Y —» Z if and only if for every directed s e t { y ...
ON CB-COMPACT, COUNTABLY CB-COMPACT AND CB
... The next scheme shows the interdependence between some compactness type properties which are de ned by open covers with sets satisfying certain properties of its boundaries: Compactness =) CB-compactness =) FB-compactness =) Clp-compactness. Sometimes it can be useful to compare CB-compactness with ...
... The next scheme shows the interdependence between some compactness type properties which are de ned by open covers with sets satisfying certain properties of its boundaries: Compactness =) CB-compactness =) FB-compactness =) Clp-compactness. Sometimes it can be useful to compare CB-compactness with ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.