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... The union of all open sets contained in A is defined to be the interior of A. Equivalently, one could define the interior of A to the be the largest open set contained in A. In this entry we denote the interior of A by int(A). Another common notation is A◦ . The exterior of A is defined as the union ...
... The union of all open sets contained in A is defined to be the interior of A. Equivalently, one could define the interior of A to the be the largest open set contained in A. In this entry we denote the interior of A by int(A). Another common notation is A◦ . The exterior of A is defined as the union ...
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... 3. Characterizations of faint θ-rare e-continuity Definition 2. A function f : X → Y is said to be faintly θ-rare e-continuous at x ∈ X if for each G ∈ θO(Y, f (x)), there exist a θrare set RG in Y with G ∩ Cl(RG ) = ∅ and U ∈ eO(X, x) such that f (U ) ⊂ G ∪ RG . If f is faintly θ-rare e-continuous ...
... 3. Characterizations of faint θ-rare e-continuity Definition 2. A function f : X → Y is said to be faintly θ-rare e-continuous at x ∈ X if for each G ∈ θO(Y, f (x)), there exist a θrare set RG in Y with G ∩ Cl(RG ) = ∅ and U ∈ eO(X, x) such that f (U ) ⊂ G ∪ RG . If f is faintly θ-rare e-continuous ...
On productively Lindelöf spaces
... γ -spaces were introduced by Gerlits and Nagy [17], who proved that, for Tychonoff spaces, X is a γ -space if and only if the space C p ( X ) (the continuous the real-valued functions on X with the topology of pointwise convergence) is Fréchet– Urysohn. This is a very strong property. It is, for exa ...
... γ -spaces were introduced by Gerlits and Nagy [17], who proved that, for Tychonoff spaces, X is a γ -space if and only if the space C p ( X ) (the continuous the real-valued functions on X with the topology of pointwise convergence) is Fréchet– Urysohn. This is a very strong property. It is, for exa ...
Diagonal Conditions in Ordered Spaces by Harold R Bennett, Texas
... Our paper is organized as follows. In Section 2 we present definitions from ordered space theory and remind the reader of a standard tree construction. In Section 3 we use the tree construction to establish certain cardinal function inequalities for ordered spaces with various diagonal conditions. I ...
... Our paper is organized as follows. In Section 2 we present definitions from ordered space theory and remind the reader of a standard tree construction. In Section 3 we use the tree construction to establish certain cardinal function inequalities for ordered spaces with various diagonal conditions. I ...
Part III Topological Spaces
... Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1. A collection of subsets τ of X is a topology if 1. ∅, X ∈ τ S Vα ∈ τ . 2. τ is closed under arbitrary unions, i.e. if Vα ∈ τ, for α ∈ I then α∈I ...
... Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1. A collection of subsets τ of X is a topology if 1. ∅, X ∈ τ S Vα ∈ τ . 2. τ is closed under arbitrary unions, i.e. if Vα ∈ τ, for α ∈ I then α∈I ...
Universitat Jaume I Departament de Matem` atiques BOUNDED SETS IN TOPOLOGICAL
... world. Without his patience, availability and generosity with his time, this research could never have been carried out. ...
... world. Without his patience, availability and generosity with his time, this research could never have been carried out. ...
Homotopy
... Two continuous functions from one topological space to another are called homotopic if one can be “continuously deformed” into the other, such a deformation being called a homotopy between the two functions. More precisely, we have the following definition. Definition 1.1. Let X, Y be topological sp ...
... Two continuous functions from one topological space to another are called homotopic if one can be “continuously deformed” into the other, such a deformation being called a homotopy between the two functions. More precisely, we have the following definition. Definition 1.1. Let X, Y be topological sp ...
Permutation Models for Set Theory
... They were found very useful in the subsequent decades for establishing nonimplications—for instance, that the ordering principle does not imply the axiom of choice—but soon fell out of use for this purpose when Paul Cohen introduced the forcing method in the 1960s. The reason for this is that permut ...
... They were found very useful in the subsequent decades for establishing nonimplications—for instance, that the ordering principle does not imply the axiom of choice—but soon fell out of use for this purpose when Paul Cohen introduced the forcing method in the 1960s. The reason for this is that permut ...
