Lecture Notes on Metric and Topological Spaces Niels Jørgen Nielsen
... means “to choose”. This means that one has to give a rule how to choose so the mathematical correct way of formulating the above is: The Axiom of Choice If {Ai | i ∈ I} is a family of non–empty sets, then there exists a function f : I → ∪i∈I Ai so that f (i) ∈ Ai for all i ∈ I. The Axiom of Choice w ...
... means “to choose”. This means that one has to give a rule how to choose so the mathematical correct way of formulating the above is: The Axiom of Choice If {Ai | i ∈ I} is a family of non–empty sets, then there exists a function f : I → ∪i∈I Ai so that f (i) ∈ Ai for all i ∈ I. The Axiom of Choice w ...
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... Take U = f −1 (F1 ) and V = f −1 (F2 ). We have U ∩ V = ∅. Since X is θ-normal, there exist disjoint θ-open sets A and B such that U ⊂ A and V ⊂ B. We obtain that F1 = f (U ) ⊂ f (A) and F2 = f (V ) ⊂ f (B) such that f (A) and f (B) are disjoint λ-open sets. Thus, Y is λ-normal. Recall that for a fu ...
... Take U = f −1 (F1 ) and V = f −1 (F2 ). We have U ∩ V = ∅. Since X is θ-normal, there exist disjoint θ-open sets A and B such that U ⊂ A and V ⊂ B. We obtain that F1 = f (U ) ⊂ f (A) and F2 = f (V ) ⊂ f (B) such that f (A) and f (B) are disjoint λ-open sets. Thus, Y is λ-normal. Recall that for a fu ...
Topologies on spaces of continuous functions
... kind [4], which apparently is expected to be published as part of collected works. The methods that we use are different, although naturally there are common ingredients. Eilenberg’s method is to consider the largest topology on the set of continuous functions for which certain probe maps are contin ...
... kind [4], which apparently is expected to be published as part of collected works. The methods that we use are different, although naturally there are common ingredients. Eilenberg’s method is to consider the largest topology on the set of continuous functions for which certain probe maps are contin ...
Extended seminorms and extended topological vector spaces
... course, · induces a topology over X, which happens to fail to be compatible with the vectorial structure of X: it is compatible with the sum but not with the scalar multiplication. In the same papers, many classical results concerning normed spaces are extended to this new framework, showing tha ...
... course, · induces a topology over X, which happens to fail to be compatible with the vectorial structure of X: it is compatible with the sum but not with the scalar multiplication. In the same papers, many classical results concerning normed spaces are extended to this new framework, showing tha ...
On bitopological paracompactness
... open cover of X: a cover U of X is a pairwise open cover if U ⊂ P 1 ∪ P2 and for i = 1, 2, U ∩ P i contains a non-empty set. If a set E is open in the space (X, Pi ), we write ‘E is (Pi )open’. Similarly we define (Pi )closed sets, (Pi )Fσ sets, (Pi )locally finite collection of sets, (Pi )paracompa ...
... open cover of X: a cover U of X is a pairwise open cover if U ⊂ P 1 ∪ P2 and for i = 1, 2, U ∩ P i contains a non-empty set. If a set E is open in the space (X, Pi ), we write ‘E is (Pi )open’. Similarly we define (Pi )closed sets, (Pi )Fσ sets, (Pi )locally finite collection of sets, (Pi )paracompa ...
MAP 341 Topology
... We call an open interval (x − δ, x + δ) the open ball of radius δ about x. With this definition, all open intervals are open. But we also have unions such as (0, 2) ∪ (5, 10); these are open too. Having defined ‘open’ in this way, we then define the term ‘closed’ as follows. Definition: A subset Q ...
... We call an open interval (x − δ, x + δ) the open ball of radius δ about x. With this definition, all open intervals are open. But we also have unions such as (0, 2) ∪ (5, 10); these are open too. Having defined ‘open’ in this way, we then define the term ‘closed’ as follows. Definition: A subset Q ...