PRODUCTIVE PROPERTIES IN TOPOLOGICAL GROUPS
... b) first countable ⇐⇒ metrizable (Birkhoff–Kakutani’s theorem); c) all σ-compact topological groups have countable cellularity (even more, they have the Knaster property). See [28] and [3, Corollary 5.4.8]); d) every topological group has a maximal group extension %G, called the Raı̆kov completion o ...
... b) first countable ⇐⇒ metrizable (Birkhoff–Kakutani’s theorem); c) all σ-compact topological groups have countable cellularity (even more, they have the Knaster property). See [28] and [3, Corollary 5.4.8]); d) every topological group has a maximal group extension %G, called the Raı̆kov completion o ...
topologies on spaces of subsets
... 5.8). Typical among these is that the function a:zA(zA(X)) —»cvf(X), which maps a collection of sets into its union, is continuous (Theorems 5.7.1 and 5.7.2). Next we study the relationships between a function/:X —> Fand the function it induces among the hyperspaces (Theorem 5.10). We conclude this ...
... 5.8). Typical among these is that the function a:zA(zA(X)) —»cvf(X), which maps a collection of sets into its union, is continuous (Theorems 5.7.1 and 5.7.2). Next we study the relationships between a function/:X —> Fand the function it induces among the hyperspaces (Theorem 5.10). We conclude this ...
Normal induced fuzzy topological spaces
... N (x0 ) ∩ {f (x) < rl} = φ. Thus, we infer that if there exists a point x0 ∈ X, f (x0 ) = r such that N (x0 ) is a subset of {x : f (x) > r} ∪ {x0 }, then the reverse inclusion is not true. Thus the property ∗ follows from the condition that the set of the form {x : f (x) < r}, which is a union of r ...
... N (x0 ) ∩ {f (x) < rl} = φ. Thus, we infer that if there exists a point x0 ∈ X, f (x0 ) = r such that N (x0 ) is a subset of {x : f (x) > r} ∪ {x0 }, then the reverse inclusion is not true. Thus the property ∗ follows from the condition that the set of the form {x : f (x) < r}, which is a union of r ...
COMPLETION FUNCTORS FOR CAUCHY SPACES
... denotes the filter generated on X by- (considered as a filter base on X). If (X, C) is a complete Cauchy space (i.e. convergence space), then it will be necessary to distinguish between a convergence subspace (a subspace in the usual convergence space sense) and a Cauchy subspace (with the meaning d ...
... denotes the filter generated on X by- (considered as a filter base on X). If (X, C) is a complete Cauchy space (i.e. convergence space), then it will be necessary to distinguish between a convergence subspace (a subspace in the usual convergence space sense) and a Cauchy subspace (with the meaning d ...