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Completeness and quasi-completeness
... The appropriate general completeness notion for topological vector spaces is quasi-completeness. There is a stronger general notion of completeness, which proves to be too strong in general. For example, the appendix shows that weak-star duals of infinite-dimensional Hilbert spaces are quasi-complet ...
... The appropriate general completeness notion for topological vector spaces is quasi-completeness. There is a stronger general notion of completeness, which proves to be too strong in general. For example, the appendix shows that weak-star duals of infinite-dimensional Hilbert spaces are quasi-complet ...
§5 Manifolds as topological spaces
... defined in terms of neighborhoods. So it is a local property. This is used in the following theorem. Theorem 5.3. Every smooth map F : M1 → M2 between manifolds is continuous. Proof. We know that smooth maps between (open domains of) Euclidean spaces are automatically continuous. This is established ...
... defined in terms of neighborhoods. So it is a local property. This is used in the following theorem. Theorem 5.3. Every smooth map F : M1 → M2 between manifolds is continuous. Proof. We know that smooth maps between (open domains of) Euclidean spaces are automatically continuous. This is established ...
IOSR Journal of Mathematics (IOSR-JM)
... Definition 2.1: An ideal I on a nonempty set X is a collection of subset of X which satisfies the following properties: (i) A ∈ I and B ⊆ A implies B ∈ I. (ii) A ∈ I and B ⊆ I implies A ∪ B ∈ I. A topological space X, τ with an ideal I on X is called ideal topological space and is denoted by X, τ, I ...
... Definition 2.1: An ideal I on a nonempty set X is a collection of subset of X which satisfies the following properties: (i) A ∈ I and B ⊆ A implies B ∈ I. (ii) A ∈ I and B ⊆ I implies A ∪ B ∈ I. A topological space X, τ with an ideal I on X is called ideal topological space and is denoted by X, τ, I ...
Aalborg Universitet A convenient category for directed homotopy Fajstrup, Lisbeth; Rosický, J.
... (1’) U A = X and U f¯i = fi for each i ∈ I and (2’) given h : X → U B with hfi = U h̄i , h̄i : Ai → B for each i ∈ I then h = U h̄ for a unique h̄ : A → B. 3.1. Example. (1) A preordered set (A, ≤) is a set A equipped with a reflexive and transitive relation ≤. It means that it satisfies the formula ...
... (1’) U A = X and U f¯i = fi for each i ∈ I and (2’) given h : X → U B with hfi = U h̄i , h̄i : Ai → B for each i ∈ I then h = U h̄ for a unique h̄ : A → B. 3.1. Example. (1) A preordered set (A, ≤) is a set A equipped with a reflexive and transitive relation ≤. It means that it satisfies the formula ...